What is the section modulus equation for a circle?

The section modulus for a circular shape is derived from the geometry of the circle and its moment of inertia. The section modulus, (S), is defined as the moment of inertia (I) divided by the distance from the centroid to the outermost fiber ((c)).

For a circle of diameter (d) or radius (a), the moment of inertia (I) about the x-axis is given by the formula:

[ I = \frac{\pi}{64}d^4 ]

Since the diameter (d) is related to the radius (a) as (d = 2a), substituting this into the moment of inertia results in:

[ I = \frac{\pi}{64}(2a)^4 = \frac{\pi}{64} \cdot 16a^4 = \frac{\pi}{4}a^4 ]

The distance (c) from the centroid to the outermost fiber for a circle is (c = a) (the radius). Therefore, the section modulus (S) can be calculated using the formula:

[ S = \frac{I}{c} = \frac{\frac{\