Structural Engineering & Statics
where the load goes
Generator

The cut-diagram generator

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.
The same beam, cut at x = 5A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier.2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs

6 essays call cut-diagram. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs Equilibrium

The free body is a choice, and choosing it well is the whole skill

Cutting a structure open is not a step in the method. It is the method — and where the cut is made decides whether the answer takes one line or twenty.

 2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs Internal forces

What a cut reveals, and why it was there all along

Cut a beam anywhere and two quantities appear on the face — a shear force and a bending moment. Nothing was applied there. They are what the material was already doing.

 4 per unit lengthshear16.0moment32.0 at x = 4.00the moment peaks exactly where the shear passes through zero Internal forces

The diagram is an integral, and that is why it can be drawn by eye

Load, shear and moment are one function and its two integrals. Once that is seen, the diagrams stop being things to calculate and become things to sketch.

 closedJ = 5.66×10⁷ mm⁴twist 0.187° over 3.0 mpeak shear stress 8.5 N/mm²slit along its lengthJ = 1.31×10⁵ mm⁴twist 80.950° over 3.0 mpeak shear stress 305.2 N/mm²J closed ÷ J open = 432 Internal forces

The moment that will not lie flat

A plane cut exposes three actions. A real cut exposes six, and the fourth of them behaves unlike the others — torsion is resisted by a loop of shear, and one slit down the length of a tube destroys it.

 span ÷ depth = 8plane sections holdspan ÷ depth = 4plane sections holdspan ÷ depth = 2off by 19%span ÷ depth = 1off by 31%the assumption is the theory — everything else is arithmetic on top of it Sections and stress

Plane sections stay plane, and what the assumption costs

Beam theory rests on one sentence about geometry. It is very nearly true for a slender member, wrong for a deep one, and everything in the subject that fails does so where it stops holding.

 2000400060008000100001200002004006008001000distance between lateral restraintsthey cross at 3803the plastic capacity of the sectionelastic critical momentSt Venant torsion alone — what is left at long lengthswarping dominates here Stability

The beam that fails sideways

A deep narrow beam bending in its strong plane can, at a moment well below its capacity, swing out of that plane and twist. The failure has nothing to do with how much it can carry and everything to do with what is holding it.

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