Why triangles hold a bridge up

A truss holds a load up with a frame that is mostly air: long slender members joined only at their ends, wired into triangles — shapes whose angles cannot change while their sides keep their length.

Members, joints, and a bargain

A truss is a rigid structure of long, slender members connected at their ends. It spans long distances with a strong, lightweight frame — bridges, roofs and pylons. Flat trusses are built from triangular subunits; space trusses use a tetrahedron.

Real members are welded or riveted through a gusset plate, and a joint that rigid makes analysis miserable. So the analysis bargains: joints become frictionless pins, and every load and reaction is applied at a pin, never along a member. Each member then feels force only at its two ends, along its own axis — a two-force member, able to pull or push along that axis and nothing else.

What the bargain buys

One unknown per member instead of a whole distribution of internal force. And a simple truss has as many equilibrium equations as unknowns: twice the number of joints, against the reactions plus the members. With the usual three reactions, 2 × joints = 3 + members — determinate, and solvable by statics alone.

The same count names the two ways a truss goes wrong. Fewer unknowns than equations: it lacks the members to hold its shape off its supports. More unknowns than equations: the extra members are redundant, and statics cannot resolve them.

Solving it one joint at a time

The method of joints works on the pins. Each joint is a particle: every member's axial force, the load and the reaction act through one point, so the forces are concurrent. Equilibrium there is ΣF = 0 and nothing else; the moment equation gives no information.

Each pin yields two scalar equations, one for x and one for y, so a joint can be solved once no more than two of its forces are unknown and at least one is known. It is bookkeeping: solve the joint with the fewest unknowns, carry its answers into its neighbours' free-body diagrams, keeping each force's sense. Assume every unknown in tension, pulling away from the pin; a negative answer is compression. The last joint then checks the work.

Members that carry nothing

Some members carry no force at all, and two rules find them by inspection. Two non-collinear members meeting at an unloaded joint are both zero-force; where three forces meet and two are collinear, the third is zero-force. Removing one often exposes another.

Zero-force is not useless. A real member has weight, a long slender one in compression buckles, and a zero-force member is often there to brace one against buckling. A bridge's load moves as a truck crosses it: one carrying nothing now may hold the shape later.

The dark side of simplicity

No redundancy is what makes a simple truss solvable — and its danger. Take one member away and the force it carried has no alternative path, so one failed component can bring the whole structure down; such trusses are sometimes called fracture-critical. Break one side of a triangle with pinned corners and the other two lose their support, so in a truss of triangles the collapse spreads. Fracture-critical bridges are being replaced, but thousands are still in service in the United States.

In short

A truss is a frame of slender members joined at their ends. Idealised as frictionless pins with loads only at those pins, each member carries one number: a tension or compression along its own axis. Triangles make the frame rigid, the equation count shows it is solvable, and the method of joints is that solution one pin at a time.