Stresses & design checks

This is a static copy of the chapter for search engines. The interactive version has animated figures, check questions, and buttons that load the example into the calculator.

Once the internal actions are known, the stress at any cross-section follows from the ordinary formulae — you just have more of them to add up than in a straight beam.

Normal stress

σ = |N|A + |M|W + |M_⊥|W_⊥

W = I/c is the section modulus for bending in the frame plane and W_⊥ the one for bending out of it. Adding the absolute values puts the peak at the worst corner of the cross-section, which is where biaxial bending really peaks. It is mildly conservative and completely safe.

When a bar bends about both axes at once the critical point moves to a corner of the cross-section. FrameLab adds the three normal-stress contributions there, which is safe (slightly conservative) for any section shape.

Which axis is which

A rectangular or I-shaped section is much stiffer about one axis than the other, so how you orient it matters. FrameLab has a switch for this: the section’s strong axis can lie in the frame plane (the normal choice — the web of an I-beam is in the plane of the frame) or out of it. Flip it and watch every result change.

Shear stress from torsion

τ = |T|W_t

with W_t the torsional section modulus: πd³/16 for a solid circle, 2·A_m·t for a thin closed section (Bredt), and J_t/t_max for an open one.

Combining them

Normal and shear stress at the same point are combined with the von Mises criterion:

σ_eq = √(σ² + 3τ²) ≤ f_yk

where k is the safety factor you choose. FrameLab evaluates this at every sampled cross-section along the bar and reports the worst one, together with the utilisation as a percentage.

What the check does not cover

Stress concentration at the corner. Nominal stresses only — see chapter 4.
Buckling. Frame columns are compression members. A slender column can buckle long before σ reaches f_y, and a deep beam can buckle laterally. Neither is checked here.
Shear stress from V. Usually small compared with bending in a slender bar, but not always — check it separately for short, deep members.
Warping normal stress in open sections under torsion (see chapter 7).
Fatigue, which for a welded corner is often the real governing criterion.

Displacement check

The tool reports the largest in-plane displacement δ = √(u² + v²) and, when relevant, the largest out-of-plane deflection w, and compares the larger of the two with a limit of S/n or an absolute value you set. S is the developed length of the whole bar — a pragmatic yardstick for a bent bar, but do think about whether the limit that matters in your application is really about the total length or about one particular member.

💡 A useful habit: check stress and displacement separately. Stiffness scales with I, strength with W = I/c. Making a section deeper helps deflection much more than it helps stress.