Shape Factor for Circular Section

The shape factor is a property of a cross-section that describes its efficiency in resisting bending moment under plastic conditions compared to elastic conditions. It is defined as the ratio of the plastic section modulus ($Z_p$) to the elastic section modulus ($Z_e$).

Shape factor $ = \frac{Z_p}{Z_e} $

Elastic Section Modulus ($Z_e$)

The elastic section modulus ($Z_e$) is given by the ratio of the moment of inertia ($I$) about the neutral axis to the distance of the extreme fiber from the neutral axis ($y_{max}$).

For a solid circular section with diameter $D$:

• Moment of inertia, $I = \frac{\pi D^4}{64}$
• Distance to extreme fiber, $y_{max} = \frac{D}{2}$

So, the elastic section modulus is:

$ Z_e = \frac{I}{y_{max}} = \frac{\frac{\pi D^4}{64}}{\frac{D}{2}} = \frac{\pi D^4}{64} \times \frac{2}{D} = \frac{2\pi D^3}{64} = \frac{\pi D^3}{32} $

Plastic Section Modulus ($Z_p$)

The plastic section modulus ($Z_p$) for a section about an axis is the sum of the first moments of area of the two parts of the cross-section divided by the plastic neutral axis. For a solid circular section, the plastic neutral axis passes through the centroid and divides the circle into two semi-circles.

For a solid circular section with diameter $D$ (radius $R=D/2$):

• Area of each semi-circle = $\frac{1}{2} \pi R^2 = \frac{1}{2} \pi \left(\frac{D}{2}\right)^2 = \frac{\pi D^2}{8}$
• Distance of the centroid of a semi-circle from the diameter (plastic neutral axis) = $\frac{4R}{3\pi} = \frac{4(D/2)}{3\pi} = \frac{2D}{3\pi}$

The plastic section modulus is the sum of the moments of area of the two semi-circles about the plastic neutral axis:

$ Z_p = \left(\text{Area of top semi-circle} \times \text{Centroid distance}\right) + \left(\text{Area of bottom semi-circle} \times \text{Centroid distance}\right) $

$ Z_p = \left(\frac{\pi D^2}{8} \times \frac{2D}{3\pi}\right) + \left(\frac{\pi D^2}{8} \times \frac{2D}{3\pi}\right) $

$ Z_p = 2 \times \left(\frac{\pi D^2}{8} \times \frac{2D}{3\pi}\right) = 2 \times \frac{2\pi D^3}{24\pi} = \frac{4\pi D^3}{24\pi} = \frac{D^3}{6} $

Alternatively, using radius $R$:

• Area of each semi-circle = $\frac{\pi R^2}{2}$
• Centroid distance = $\frac{4R}{3\pi}$

$ Z_p = 2 \times \left(\frac{\pi R^2}{2} \times \frac{4R}{3\pi}\right) = 2 \times \frac{4\pi R^3}{6\pi} = \frac{8\pi R^3}{6\pi} = \frac{4R^3}{3} $

Substituting $R = D/2$:

$ Z_p = \frac{4(D/2)^3}{3} = \frac{4(D^3/8)}{3} = \frac{D^3/2}{3} = \frac{D^3}{6} $

Both methods give the same result for $Z_p$.

Calculating Shape Factor

Now, we calculate the shape factor using the values of $Z_p$ and $Z_e$:

$ \text{Shape factor} = \frac{Z_p}{Z_e} = \frac{\frac{D^3}{6}}{\frac{\pi D^3}{32}} $

$ \text{Shape factor} = \frac{D^3}{6} \times \frac{32}{\pi D^3} = \frac{32}{6\pi} = \frac{16}{3\pi} $

Calculating the numerical value:

$ \frac{16}{3\pi} \approx \frac{16}{3 \times 3.14159} \approx \frac{16}{9.42477} \approx 1.6976 $

This value is approximately $1.7$.

Summary of Section Properties

Therefore, the shape factor for a circular section is approximately 1.7.