Area Moment of Inertia: Definition, Formula, Theorem

Area Moment of Inertia

The area moment of inertia, also known as the second area moment or the 2nd moment of area, is a feature of a two-dimensional plane form that illustrates how its points are distributed in the cross-sectional plane along an arbitrary axis. This property describes the deflection of a plane shape under a load. The area moment of inertia is essential for the GATE exam as MCQ-based questions can be seen in the exam.

For an axis in a plane, the area moment of inertia is commonly indicated by the symbol I. When the axis is perpendicular to the plane, it is also designated as J. The dimension unit of the second area moment is L⁴ (length to the power of four). The International System of Units unit of measurement is the meter to the power of four, or m⁴. It can be inches to the fourth power, in⁴ if we use the Imperial System of Units.

This concept will frequently appear in the field of engineering. In this context, the area moment of inertia is believed to measure a beam's flexural stiffness. It is a significant feature used to calculate a beam's deflection or measure its resistance to bending. We'll look at two examples here.

• First, the planar second moment of the area where the force sits perpendicular to the neutral axis can characterize or define a beam's bending resistance.
• Second, when the applied moment is parallel to the beam's cross-section, the polar second moment of the area can be employed to calculate its resistance. It is essentially the beam's resistance to torsion.

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Area Moment of Inertia Formulas

The area moment of inertia formulas can be used to formulate the MSQ-based questions carrying marks scheduled as per the GATE exam pattern. When we consider the area moment of inertia for the x-axis, we get;

I_x = I_xx = ∫y²dxdy

Meanwhile, the area's "product" moment is described by

I_xy = ∫xy dx dy

Parallel Axis Theorem

The parallel axis theorem is an essential part of the GATE ME syllabus. A body's moment of inertia about any axis is equal to the sum of its moment of inertia about a parallel axis passing through its center of mass, the product of its mass, and the square of the distance between the two lines.

I = I_g = Md²

Where:

• M = Mass of the body
• D = the perpendicular distance between the two lines

Perpendicular Axis Theorem

The moment of inertia about a perpendicular axis for a planar body is the sum of the moments of inertia about two perpendicular axes coincident with this perpendicular axis. It sits on the plane of the body. The expression specifies it.

I_z = I_x + I_y

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Area Moments of Inertia for Some Common Shapes

The following is a list of some form's second moments of area. The second moment of area, also known as the area moment of inertia, is a geometrical feature of an area that represents the distribution of its points concerning an arbitrary axis.

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