Normal Stress in Simple Bending — Navier Formula

For simple bending of a beam within the linear-elastic model, normal stress varies linearly with distance from the neutral axis:

$$\sigma_x(y)=-\frac{My}{I}.$$

The maximum absolute stress at an extreme fiber is:

$$|\sigma_{\max}|=\frac{|M|y_{\max}}{I}=\frac{|M|}{W},\qquad W=\frac{I}{y_{\max}}.$$

• $M$ — bending moment;
• $I$ — second moment of area about the neutral axis;
• $y$ — distance from the neutral axis;
• $y_{\max}$ — distance to the relevant extreme fiber;
• $W$ — section modulus.

The sign of $\sigma_x$ depends on the sign convention for $M$ and $y$; one side of the section is in tension and the other in compression.