Structural Engineering & Statics
where the load goes
Concept

Rank — where it appears

The number of independent equations in a set, which decides whether a structure stands rather than the number of equations there are. A frame whose equilibrium matrix has full rank is stable whatever the member count says, and Maxwell's rule is a corollary of the rank rather than a substitute for it.

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

Three legs, three equations, one answer. A rigid top on three legs carrying 120 kN at (0.3, 0.2) m. The three equilibrium equations available — one vertical and two moments — leave three unknowns, so the system is exactly determinate and the reactions are 56.8, 13.6, 49.6 kN. Move the load anywhere and the answer moves with it; nothing about the legs' stiffness enters.

Six equations, and the drawing shows three

Every essay so far has taken place on a piece of paper, where equilibrium is three equations and a structure is a diagram. The real object has six, the extra three are the ones nobody writes down, and the difference between three legs and four is not a matter of degree.

equilibrium · Three-dimensional equilibrium Two null spaces of one matrix, and the count is their difference. Two pin-jointed frames, each with the forces it can carry with nothing applied to it drawn on its bars — tension one colour, compression the other, thickness in proportion. That force set is the null space of the equilibrium matrix; a mechanism is the null space of its transpose; and Maxwell's count b + r − 2j is the difference of their dimensions and knows neither of them separately. A square with both diagonals has s = 1 and m = 0. Two bars in a straight line has s = 1 and m = 1 with a count of 0, so the count is satisfied by a frame that both folds and can be prestressed — and the prestress stiffness is positive, which is why a tensioned pair of collinear bars is stiff at all.

The forces that are there with nothing applied

Maxwell's count is the difference between two dimensions, and it knows neither of them separately. A frame can satisfy it exactly and still both fold and be prestressable — and when it does, the second of those is what stops the first.

structures · Self-stress

Named alongside it

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

DeterminacySelf-stressCable netEquilibrium matrixFree bodyGeometric stiffnessIndeterminacyInfinitesimal mechanismMaxwell countMechanismMoment about an axisNull space

All concepts