What happens at a corner

This is the heart of the subject. Take a rigid corner with no load applied exactly on it, cut the bar immediately before and immediately after, and write equilibrium for the (massless, infinitesimal) piece of corner in between.

The three rules

Let Δα = α₂ − α₁ be the kink angle from member 1 to member 2. Resolving member 1’s internal force into member 2’s local axes gives:

The rotation turns the other way than you might expect, because a positive V acts along −n on a +t face: the transmitted force vector is N·t − V·n, and it is that vector which rotates. If the signs ever confuse you, read the magnitudes off the diagram drawn on the structure — that picture never depends on a convention.

Three consequences worth memorising:

1. M is continuous. A rigid unloaded corner never changes the bending moment. If your hand calculation shows a jump in M at a corner, you have either applied a moment there or made a mistake.
2. N and V rotate into each other. At a right-angle corner the swap is total: |N₂| = |V₁| and |V₂| = |N₁|, with the signs following the direction of the turn (Δα = −90° gives N₂ = V₁, V₂ = −N₁; Δα = +90° gives N₂ = −V₁, V₂ = N₁). The shear that the beam delivers to the corner leaves as pure axial force in the column.
3. Nothing is lost. The corner is a transmitter, not a source. Whatever arrives, leaves — just re-labelled.

Why the diagrams jump

Plotted against the developed length s, N(s) and V(s) show a step at every corner while M(s) runs through smoothly. That step is not a numerical artefact; it is rule 2 above. When you see it on the chart, you are looking at the geometry doing its work.

The corner is also the most stressed point

In almost every bent bar, the largest bending moment sits at a corner — because the corner is where the lever arm from the load is longest and where two members hand their moments to each other. Add to that:

• the corner usually carries the largest axial force too, so σ = N/A + M/W peaks there;
• real corners have welds, notches or bends, all of which raise the local stress well above the beam value;
• the beam theory that FrameLab uses assumes plane sections stay plane, which is not true within roughly one section depth of a sharp corner.