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Force –System Resultants -- Moments

Key Concept: If the the forces on a body or in a system (ΣF_i) sum to F_r and also if

the moments of all forces about a designated point, P, and the couples (double-headed

arrows) (ΣM_Pi) on a body or in a system sum to M_rP, then the force-couple pair F_r

and M_rPis called the resultant of the system.

In a Nut Shell:

In general any force system, including couples acting on a body, can be replaced by a resultant

force F_r and a resultant moment, M_rP. An equipollent (or equivalent) force system means that

we may replace any system, such as system 1 shown below with a resultant force F_r at P and a

resultant couple M_r as shown in system 2 below.

If when summing moments in system 1 point P is selected for all the forces in system 1,

then the moment of these forces plus the moment of all couples in system 1 must add up

to M_rP in system 2 at point P.

Note: Although F_r does not depend on the choice of P M_rP depends on the line of

action of F_r.

Σ F_i = F_r = Resultant Force in system 2

Σ Couples + Σ r_Pi x F_i = M_rP = Resultant Moment in system 2

where F_i are all the forces acting on system 1, r_Pi are the vectors from P to any point

on the line of action for forces F_i in system 1. Note that couples are independent of

the point about which moments are summed.