Resultants of force systems
Many forces acting on a body can be replaced by a single resultant force and, sometimes, a couple. This simplification tells engineers the net effect of complicated loads.
Resultant force and moment
Any system of forces and couples can be reduced to a resultant force \(\mathbf{F}_R = \sum \mathbf{F}\) at a chosen point O and a resultant couple \(\mathbf{M}_{R_O} = \sum \mathbf{M}_O\), the sum of the moments of all forces about O plus any couples. Different choices of O change the couple but not the force.
Parallel forces
For parallel forces, such as loads on a beam, the resultant is their sum, acting at a location where its moment equals the total moment. Loads of 200 N at 1 m, 300 N at 3 m, and 500 N at 6 m from the left end total 1,000 N, with a moment of 200 + 900 + 3,000 = 4,100 N·m about the left end, so the resultant acts at 4.1 m.
Coplanar systems
For forces in one plane, the force-couple system can always be reduced further to a single force, placed so that its moment about O equals \(M_{R_O}\), unless the resultant force is zero. If \(\sum \mathbf{F} = 0\) but the moments do not cancel, the system reduces to a pure couple.
Locating the resultant line
In 2D, if the resultant has components \(R_x\) and \(R_y\) and the total moment about O is M, the line of action satisfies \(M = xR_y - yR_x\). Setting y = 0 gives where the resultant crosses the x-axis, often the most useful reference point.
Concurrent systems
Forces that all pass through one point reduce to a single force through that point, with no couple, since each has zero moment about the common point. This is why particle equilibrium needs only force equations.
Three dimensions and wrenches
In 3D, a general system reduces to a resultant force and a couple, which can be simplified to a wrench: a force plus a couple parallel to it, like the push and twist on a screwdriver. Only special systems, such as concurrent, coplanar, or parallel ones, reduce to a single force.
Practical uses
The resultant of all the loads on a building's floor tells engineers the total weight and where it effectively acts, which determines how load is shared between columns. Ship designers locate the resultant of weight and buoyancy to check stability.
Worked example
A 10 m beam carries 20 kN at 2 m, 30 kN at 5 m, and 10 kN at 9 m, all downward. The resultant is 60 kN. Moments about the left end: 40 + 150 + 90 = 280 kN·m, so the resultant acts at 280 / 60, about 4.67 m. A single 60 kN load at 4.67 m produces the same support reactions as the three loads.
Key takeaways
Any force system reduces to a resultant force, the vector sum, and a resultant couple, the total moment about a chosen point. Parallel forces have a resultant equal to their sum, located where its moment matches the total moment. Coplanar systems reduce to a single force unless the forces sum to zero, leaving a pure couple. Concurrent forces reduce to one force through their common point. General 3D systems reduce to a wrench. Resultants determine how loads are shared by supports.
Flashcard
Vector sum of all forces
Check your understanding
Downward loads of 200 N at 1 m, 300 N at 3 m, and 500 N at 6 m act on a beam. Where does the resultant act?