Trusses by joints and sections

A truss is a framework of straight members joined at their ends to form triangles. Because a triangle cannot change shape without a member changing length, a truss is rigid and efficient. In the ideal model, joints are frictionless pins and loads are applied only at joints, so every member carries pure axial force: tension or compression, with no bending. Roof trusses, bridge girders, transmission towers and crane booms are all analysed this way first.

A plane truss is statically determinate when m + r = 2j, where m is the number of members, r the number of reaction components and j the number of joints. If m + r is less, it is a mechanism; if more, it is indeterminate and needs deformation methods.

The method of joints isolates each pin. Two equilibrium equations (ΣFx = 0, ΣFy = 0) are available per joint, so start at a joint with no more than two unknown members, usually a support after finding reactions. Work joint to joint. Assume every unknown member is in tension (pulling away from the joint); a negative result means compression. Watch for zero-force members: at an unloaded joint with two non-collinear members, both are zero; at an unloaded joint where two members are collinear, the third is zero.

The method of sections cuts the truss through no more than three members of unknown force and treats one side as a free body with three equations. Taking moments about the point where two of the cut members meet gives the third directly. It is the fastest way to get one member force in the middle of a long truss, and it underlies the influence lines used for moving loads on bridge trusses. Real trusses have rigid joints and self-weight, causing small secondary bending, which design codes deal with separately.