Structural Engineering & Statics
where the load goes
Concept

Independent equations — where it appears

The equations in a set that are not combinations of the others, whose number rather than whose total decides what can be solved. A structure whose equilibrium equations are dependent is a mechanism however many of them there are.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A beam, its loads and its reactions. A free body: the beam cut away from everything it touches, with the forces that were doing the touching drawn on it. The reactions are computed from the loads, so their arrows are to scale relative to each other. The moment of everything on the body is summed about three marked points, and every sum comes to zero. Nothing here is a new equation: with only vertical forces on the body, the moment about a point does not depend on that point's height, so every centre anywhere on the plane returns the same equation — and the third equilibrium equation, the horizontal sum, reads nothing equals nothing.

The equation that is not new, and the three that are

A plane free body yields exactly three independent equations. Most attempts at a fourth are one of the first three wearing different clothes — and on a beam under vertical load, one of the three is already saying nothing.

equilibrium · Equilibrium The count is necessary and not sufficient. Two pin-jointed frames, each satisfying m + r = 2j exactly. One of them folds anyway, because the equations are not independent; the ghosted outline is the motion that costs no member any change of length, drawn at an exaggeration of 0.55 of the span.

The count that does not see it

A frame can have exactly as many unknowns as equations and fold up anyway. The count asks whether there are enough equations; it never asks whether they are different from one another.

equilibrium · Determinacy

Named alongside it

The objects these essays reach for when they reach for this one.

BracingCritical formDeterminacyEquilibrium matrixForce equilibriumFree body diagramLine of actionMechanismMoment centreMoment equilibriumRank deficiencyRigid body

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