Euler Beam Theory

1 Euler Beam Theory of Pure Bending

Bending stress:

\sigma_z = E\kappa y

2 Geometry of the Cross-Section

The position of the centroid y_G is given by y_G = \frac{\int_A y' dA}{\int_A dA}

The second moment of area is given by I_{xx} = \int_A y^2 dA

This tells you the distribution of area with respect to the centroid.

Definition 1 (Neutral Axis) The neutral axis of a beam is the line along which there is no tension or compression in the fibers of the beam.

3 Relationship between Bending Stress and Bending Moment

\left. \begin{matrix} \sigma &= E \kappa y \\ M &= EI\kappa \end{matrix}\, \right\}

Therefore \sigma = \frac{y}{i}M \textnormal{or} M = \frac{I}{y}\sigma

from which we create the section modulus Z_t = I/y.