Varignon's theorem
A theorem stating that the torque of a resultant of two concurrent forces about any point is equal to the algebraic sum of the torques of its components about the same point.
Varignon's theorem: how force couples break into parts
Varignon's theorem describes a fundamental property of force systems: when two or more forces meet at a point, the moment (torque) produced by their resultant around any other point equals the sum of the moments each force would produce independently around that same point. In practical terms, this means you can analyze a complex load as a collection of simpler components without losing accuracy about its rotational effect on a structure.
The theorem applies wherever concurrent forces meet. Consider a bracket or pin joint carrying multiple loads from different directions. Rather than first finding the resultant force and then calculating its moment, you can calculate the moment of each individual force and add them algebraically. Both methods yield identical results. This flexibility makes the theorem invaluable in stress analysis, particularly when forces arrive at awkward angles or when the geometry makes direct resultant calculation tedious.
Why this matters in practice
Machine designers use Varignon's theorem to check whether a fastener or weld can safely carry combined loads. A shaft under both bending and torsion, for example, can be treated as the superposition of separate moment components. Structural analysts apply it when verifying that a joint can handle moments from multiple attached members. In hydraulic systems, force moments from angled cylinders are decomposed and summed. The theorem eliminates the need to construct resultant force polygons by hand, a method that is both time consuming and prone to drawing error.
The theorem's power lies in its linearity: moments add like scalars along the same axis. This is why sign convention matters. Forces turning counterclockwise around a point contribute positive moment; clockwise forces contribute negative moment. If moments cancel, the net rotational effect is zero, even if individual forces are substantial. Misapplying signs is the most common error in application.
Named after the French mathematician Pierre Varignon (1654-1722), the theorem is sometimes called the moment of a resultant. It is a cornerstone of statics and appears in every mechanics text, yet its name is rarely used outside academic settings; engineers simply apply it implicitly when decomposing loads. It underpins the method of sections in truss analysis and the moment distribution method in frame analysis.