1. Bending Stress ($\sigma$)

When a beam bends, the top fibers are in compression and the bottom fibers are in tension (or vice versa). The maximum stress occurs at the furthest distance ($c$) from the Neutral Axis.

Where $M$ is the Bending Moment and $I$ is the Moment of Inertia. To reduce stress, increase the depth of the beam (which increases $I$ cubically) rather than the width.

2. Shear Stress ($\tau$)

Shear force ($V$) tries to slice the beam vertically. The average shear stress is simply $V/Area$. However, for rectangular beams, the peak shear stress at the neutral axis is 1.5x the average ($1.5 V/A$). For I-beams, the web takes almost all the shear load.

3. Deflection Limits ($\delta$)

A beam might be strong enough not to break (Stress < Yield), but too flexible for use. Excessive deflection causes plaster to crack, machinery to misalign, or puddling on roofs.

Standard Limits:
Structural Floor: $L/360$
Roof Supporting Ceiling: $L/240$
Industrial Piping: $10\text{mm}$ or $L/500$

Deflection is controlled by Stiffness ($EI$). $E$ depends on material (Steel is 3x stiffer than Aluminum), and $I$ depends on geometry.

4. SFD & BMD Diagrams

These diagrams are the roadmap of beam forces.
Shear Force Diagram (SFD): Shows vertical force at every point. Value changes at point loads. Slope changes at UDLs.
Bending Moment Diagram (BMD): Integral of Shear. Shows where the beam wants to bend most. Max Moment occurs where Shear crosses zero.