Structural Engineering & Statics
where the load goes
Generator

Three legs, three equations, one answer

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.
Three legs, three equations, one answer. A rigid top on three legs carrying 100 kN at (0.4, 0.25) m. The three equilibrium equations available — one vertical and two moments — leave three unknowns, so the system is exactly determinate and the reactions are 60.4, 3.1, 36.5 kN. Move the load anywhere and the answer moves with it; nothing about the legs' stiffness enters.

Three legs, three equations, one answer. A rigid top on three legs carrying 100 kN at (0.4, 0.25) m. The three equilibrium equations available — one vertical and two moments — leave three unknowns, so the system is exactly determinate and the reactions are 60.4, 3.1, 36.5 kN. Move the load anywhere and the answer moves with it; nothing about the legs' stiffness enters.

3 essays call space-equilibrium. 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.

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. Equilibrium

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.

 A frame with no plane to be drawn in. The tetrahedron, solved: three equations at every free joint, one axial force in every member. Members drawn heavy carry more; tension and compression are separated by the sign that came back from the solve rather than by inspection. The count is m + r = 12 against 3j = 12, which makes it exactly determinate, and joint equilibrium closes to 1.8e-15. Structural form

Three equations at every joint

A plane truss is determinate when m + r = 2j. A space frame needs 3j, and that one changed digit is why a cube of twelve bars is six mechanisms short while looking perfectly solid — and why every three-dimensional frame ever built is made of triangles in several planes at once.

 At thirty degrees each leg carries the whole load. A 100 kN lift on two legs at 60 degrees to the horizontal. Each leg is a two-force member, so its force is the vertical share it carries divided by the sine of its angle: T = W/(2 sin β) = 57.7 kN, which is 0.58 times the whole load in each. The curve on the right is that division, and it is the reason the rule about sling angles exists rather than a convention: at sixty degrees a leg carries 0.58 W, at forty-five 0.71, at thirty exactly 1.00, and at fifteen 1.93. And the horizontal components do not disappear — they run through the thing being lifted, which here carries 57.7 kN of compression between the two pick points. That is what a spreader beam is for: it takes the compression as a designed strut so the legs above it can stand up. Equilibrium

The angle that doubles the force

A crane hook takes the weight and nothing else. What holds the load is a set of legs running down to it at an angle, and each leg is a two-force member — so at thirty degrees from the horizontal each of two legs carries the entire load, and the difference goes into the thing being lifted as compression.

The library, page 5 of 7 — where space-equilibrium sits