Determinacy and stability

Before analysing a structure we check two things. Is it stable, meaning it cannot move as a mechanism? And is it statically determinate, meaning statics alone gives all the forces?

For a plane truss with m members, r reaction components and j joints, the count m + r compared with 2j is the first check. If m + r is less than 2j the truss is a mechanism. If it is equal, it may be determinate. If it is greater, it is indeterminate to the degree of the difference. The count is necessary but not sufficient: members arranged badly can leave a part free to move even when the numbers look fine, so the geometry must also be checked.

For plane frames, each member adds three independent internal force unknowns (its end forces, less the three equations of its own equilibrium), each reaction component adds one, and each rigid joint gives three equations. A common shortcut counts the degree of indeterminacy as 3m + r − 3j for rigid-jointed plane frames, reduced by any releases such as internal hinges.

Indeterminate structures are not bad. They are usually stiffer and can redistribute load if one part yields, which is valuable in earthquakes. They simply need compatibility equations, which is why slope-deflection, moment distribution and the stiffness method exist.