Design considerations for prismatic beams

A prismatic beam has a constant cross-section along its length. Design means choosing a section that keeps the maximum stresses within allowable limits and, where needed, the deflection within the permitted value.

Basic considerations

1. Bending governs most designs. The required section modulus is

$$Z_{req} = \frac{M_{max}}{\sigma_{all}}$$

2. Shear must be checked, especially for short, heavily loaded beams, and for thin webs: $\tau_{max} \le \tau_{all}$.
3. Principal stresses may be critical at points where both $\sigma$ and $\tau$ are large, such as the web-flange junction of an I-beam.
4. Deflection limits (for example span/325 to span/250 for many structural members) and stability against lateral buckling may govern.
5. Economy: choose the section with the least area satisfying $Z \ge Z_{req}$, with depth $>$ width so that the material is placed away from the neutral axis.

Ductile versus brittle materials

Short versus long beams

• Long beams ($L/d$ greater than about 10 to 15): bending governs; the shear stress is small.
• Short beams ($L/d$ small): the shear force is large relative to the moment; the shear stress, bearing at supports and web crippling may govern.

Design steps

1. Find reactions, draw the shear force and bending moment diagrams, get $M_{max}$ and $V_{max}$.
2. Compute $Z_{req} = M_{max}/\sigma_{all}$.
3. Select a section (standard section from tables or fix the proportions $h = 2b$ for timber).
4. Check shear stress at the NA, principal stresses at critical points, and deflection.
5. Revise the section if any check fails.