Summary of Key Concepts

  1. An axis through the cross-section is the natural axis of bending if: $$ \int y \, dA = 0 $$ This condition ensures the axis passes through the centroid.
  2. The flexure formula relates bending stress \( \sigma_y \), bending moment \( M \), and curvature \( \rho \): $$ \frac{\sigma_y}{y} = \frac{M}{I} = \frac{E}{\rho} $$

Theorems for Moment of Inertia

1. Parallel Axis Theorem

The moment of inertia \( I_{xx} \) about any axis \( xx \) is related to the moment of inertia about a parallel axis through the centroid \( I \), and the perpendicular distance \( d \) between the axes:

$$ I_{xx} = I + A d^2 $$
  • \( I \): Moment of inertia about centroidal axis
  • \( A \): Area of the section
  • \( d \): Distance between centroidal axis and new axis

2. Perpendicular Axis Theorem

For a planar area lying in the \( xy \)-plane, the moment of inertia about the axis perpendicular to the plane (i.e., \( z \)-axis) is:

$$ I_z = I_x + I_y $$

This theorem is applicable only to flat, planar sections.

Applications

  • Locating the neutral axis in bending problems
  • Calculating section modulus for strength design
  • Analyzing composite and built-up sections
  • Deriving expressions for standard shapes (rectangles, circles, I-beams)