Hollow Circular Column Bending Moment of Inertia (Civil Engineering) Solution

A civil and structural bridge engineer at the UNLV Howard R. Hughes College of Engineering is designing hollow reinforced concrete pier columns to support an elevated multi-lane viaduct over the Las Vegas Freeway (I-15).

The column cross-section is fabricated as a thick-walled hollow circular ring (annulus) with an inner radius of $R_1 = 1\text{ meter}$ and an outer radius of $R_2 = 2\text{ meters}$. Its structural resistance against bending about the horizontal neutral centroidal axis (the $x$-axis) is governed by the second moment of area:

Step-by-Step Solution:

1. Express Annular Geometry as a Polar Rectangle:
   State the Annulus Domain Definition in Polar Coordinates (defined in Polar Rectangles and Annular Sectors):

   $$R = \{(r, \theta) \in \mathbb{R}^2 \mid R_1 \le r \le R_2, \, 0 \le \theta \le 2\pi\}$$
   Substitute the column physical dimensions:
   $$1 \le r \le 2\text{ m}, \quad 0 \le \theta \le 2\pi\text{ rad}$$

2. Transform Integrand and Area Differential:
   Apply the Polar Coordinate Substitution Equation:

   $$y = r \sin \theta \implies y^2 = r^2 \sin^2 \theta$$
   State the Polar Area Element Formula with Jacobian:
   $$dA = r \, dr \, d\theta$$
   Assemble the differential area moment:
   $$dI_x = y^2 \, dA = (r^2 \sin^2 \theta)(r \, dr \, d\theta) = r^3 \sin^2 \theta \, dr \, d\theta$$
   Formulate the double integral:
   $$I_x = \iint_R r^3 \sin^2 \theta \, dr \, d\theta = \int_{0}^{2\pi} \int_{1}^{2} r^3 \sin^2 \theta \, dr \, d\theta$$

3. Evaluate via Fubini's Product Separability:
   Apply Fubini's Separability Theorem in Polar Coordinates (defined in Separability in Polar Integrals):

   $$I_x = \left( \int_{0}^{2\pi} \sin^2 \theta \, d\theta \right) \left( \int_{1}^{2} r^3 \, dr \right)$$

   Evaluate the angular integral using the half-angle identity $\sin^2\theta = \frac{1 - \cos(2\theta)}{2}$:

   $$\int_{0}^{2\pi} \sin^2 \theta \, d\theta = \int_{0}^{2\pi} \frac{1 - \cos(2\theta)}{2} \, d\theta = \left[ \frac{\theta}{2} - \frac{\sin(2\theta)}{4} \right]_{0}^{2\pi} = \left(\frac{2\pi}{2} - 0\right) - 0 = \pi\text{ rad}$$

   Evaluate the radial integral:

   $$\int_{1}^{2} r^3 \, dr = \left[ \frac{r^4}{4} \right]_{1}^{2} = \frac{(2)^4}{4} - \frac{(1)^4}{4} = \frac{16 - 1}{4} = \frac{15}{4}\text{ m}^4$$

   Multiply the two independent factors:

   $$I_x = \pi \times \frac{15}{4} = \frac{15\pi}{4}\text{ m}^4 \approx 11.78097\text{ m}^4$$

Meaning of the Answer & Real-Life Application:

• Quantitative Interpretation: The second moment of area of the hollow pier column about its horizontal bending axis is exactly $\frac{15\pi}{4}\text{ m}^4 \approx 11.78\text{ m}^4$.
• Engineering Application: In civil infrastructure and bridge engineering, $I_x$ quantifies a cross-section's geometric stiffness and efficiency against bending deformation. For a hollow circular section, classical solid mechanics gives $I_x = \frac{\pi}{4}(R_2^4 - R_1^4) = \frac{\pi}{4}(16 - 1) = \frac{15\pi}{4}\text{ m}^4$, perfectly confirming the polar double integral evaluation. Structural engineers use this value in the Euler buckling equation $P_{\text{cr}} = \frac{\pi^2 E I}{L_{\text{eff}}^2}$ to verify that freeway overpass columns will not undergo elastic instability or lateral plastic hinge collapse during peak seismic ground motions.