Moments and couples

A moment is the turning effect of a force. Push a door near its hinge and it barely moves; push the same amount at the handle and it swings open. The difference is the lever arm, the perpendicular distance from the turning point to the line of action of the force. Moment equals force times that perpendicular distance, and it is measured in kilonewton-metres (kNm) or newton-millimetres (Nmm). The direction matters, so we give each moment a sign: commonly clockwise positive or anticlockwise positive, chosen once and kept for the whole problem.

Varignon's theorem says the moment of a force about a point equals the sum of the moments of its components about the same point. This is a time-saver. An inclined force on a bracket can be split into horizontal and vertical parts, each with an easy lever arm, instead of hunting for the perpendicular distance to the sloping line.

A couple is a pair of equal, opposite, parallel forces that do not share a line. Their resultant force is zero, yet they produce a pure turning effect equal to one force times the distance between them. A couple has the same moment about every point in the plane, which is why a fixed-end moment on a beam or the torque from a spanner can be drawn as a curved arrow without a location. Any force can also be moved to a new point if a couple equal to the force times the shift is added, a trick used constantly for eccentric loads on columns and footings.

At the advanced level, moments become vectors: the cross product of the position vector and the force. In three dimensions a structure must satisfy moment balance about three axes, giving six equilibrium equations in total. The same ideas reappear as bending moment in beams, overturning moment on retaining walls and dams, and torsion in shafts and curved bridge girders.