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Equivalent Force Systems Example

Medium DifficultyFE Statics

Replace a single force acting at one point with an equivalent force-couple system at a different point.

Concept

Two force systems are equivalent if they produce the same resultant force and the same resultant moment about any point. To move a force F from point A to point B without changing the effect on the rigid body: place the same force F at B and add a couple (moment) of magnitude M = F \times d, where d is the perpendicular distance between the line of action of F and point B. The sense of the couple is determined so that the moment about B is consistent. F^{'} = F (same magnitude and direction) M = F \times d (couple moment = force \times perpendicular distance) Notation: F^{'} = force at new point, M = couple moment, d = perpendicular distance.

A horizontal force F = 50 kN acts at point A on a rigid body. Find the equivalent force-couple system at point B, where B is 2 m directly below A (so the line of action of F is 2 m from B). Find: The equivalent force at B and the couple moment (magnitude and sense).

F = 50 kN (horizontal, at A) Perpendicular distance from line of action of F to B: d = 2 m

The equivalent force at B has the same magnitude and direction as F. F^{'} = F (same magnitude and direction) Force at B: 50 kN horizontal (same direction as F).

M = F \times d (couple moment = force \times perpendicular distance) M = 50 \times 2 = 100 kN \cdot m (clockwise or counterclockwise by choice) Sense: e.g. clockwise when viewed from the side, so that the moment of the original F about B is replaced by the couple.

Equivalent system at B: force 50 kN (horizontal, same as F) plus a couple of magnitude 100 kN\cdotm.

Force at new point + couple M = F \cdot d to preserve equivalence