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Sunday, 19 February 2017

WHAT IS AREA MOMENT OF INERTIA?

We were discussing “The difference between centroid and centre of gravity”, “Relationship between young modulus of rigidity and Poisson's ratio” and “Elongation of uniformly tapering rectangular rod” and we have also 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 area moment of inertia with the help of this post.

Let us first see here the basic concept of area moment of inertia

Area moment of inertia is basically defined as the sum of second moment of area of individual sections about an 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 area moment of inertia for this 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

Total area of the lamina = Sum of all small elemental areas
A = a1 + a2 + a3 + a4 + ----------

Moment of area about the OY axis will be determined by multiplying the area A with the perpendicular distance between the C.G and axis OY i.e. X.

Moment of area about the OY axis = A .X

Let us determine the moments of all small elemental areas about the OY axis and we will have following equation.

Moments of all small elemental areas about the OY axis = a1.x1 + a2.x2 + a3.x3+ a4.x4 +…….

Above equation will be termed as the first moment of area about the OY axis and it is used to determine the centroid or centre of gravity of the area of lamina.

If the first moment of area will again multiplied with the vertical distance between the C.G and axis OY, then we will have second moment of area i.e. A.X2.

By taking the definition of area moment of inertia in to consideration, we can write the equation for the sum of second moment of area of all small elemental areas about the OY axis and we will have following equation.

Second moments of all small elemental areas about the OY axis = a1.x12 + a2.x22 + a3.x32+ a4.x42 +…….
Iyy = a1.x12 + a2.x22 + a3.x32+ a4.x42 +…….
Iyy = Σ a.x2

Similarly, we will determine the sum of second moment of area of all small elemental areas about the OX axis and we will have following equation.
Ixx = a1.y12 + a2.y22 + a3.y32+ a4.y42 +…….
Ixx = Σ a.y2

Therefore, we can say here that area moment of inertia about an axis could be calculated by taking the product of area and the square of the perpendicular distance between the centre of gravity of the area and that axis.

Therefore, we have following formula for determining the area moment of inertia around the X axis and Y axis and it is as mentioned here.

Ixx = Σ a.y2

Iyy = Σ a.x2

Do you have any suggestions or any amendment required in this post? Please write in comment box.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

We will see another important topic i.e. Basic concept of radius of gyration in the category of strength of material in our next post.

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