Reduction of a Force System to a Given Point

Reduction of a force system to a specified point replaces a general set of forces and couples by a simpler equivalent system: one force applied at the selected point and one couple moment. This representation is fundamental for subsequent equilibrium analysis.

Moving a force to a specified point

A force $\vec F$ applied at point $A$ can be moved parallel to itself to point $O$ if a couple with moment $\vec M_O=\vec r\times\vec F$ is added at the same time, where $\vec r=\overrightarrow{OA}$. The resulting force-and-couple system is equivalent to the original force.

Resultant force vector and resultant moment

A general force system acting on a rigid body can be reduced to a specified point $O$ as a resultant force vector and a resultant moment:

$$\vec R=\sum_i\vec F_i,$$

$$\vec M_O=\sum_i(\vec r_i\times\vec F_i)+\sum_j\vec M_j,$$

where $\vec r_i$ is the position vector from $O$ to the point of application of $\vec F_i$, and $\vec M_j$ are applied couple moments. The resultant force vector is independent of the reduction point, whereas the resultant moment generally depends on it.

For a planar problem, the resultant force vector is described by $R_x=\sum F_{ix}$ and $R_y=\sum F_{iy}$, while the resultant moment about $O$ is the algebraic sum of the moments of all forces and applied couples.

Changing the reduction point

If a system has been reduced to point $O$, moving the reduction point to $A$ does not change the resultant force vector. The resultant moment transforms according to

$$\vec M_A=\vec M_O+\overrightarrow{AO}\times\vec R.$$

Thus, the same force system has the same resultant force vector at every reduction point, but generally different resultant moments when $\vec R\ne0$.

Main cases after reduction

• if $\vec R=0$ and $\vec M_O=0$, the system is balanced;
• if $\vec R=0$ but $\vec M_O\ne0$, the system is equivalent to a force couple;
• if $\vec R\ne0$, whether the system can be reduced further to a single resultant force depends on the relation between the resultant force vector and resultant moment.

Planar example

Suppose a 10 kN force acts in the $+y$ direction at a point 2 m to the right of $O$. When the force is moved to $O$, an additional moment $M_O=10\cdot2=20$ kN·m must be introduced. With the usual planar sign convention, this moment is positive because the original force tends to rotate the body counterclockwise about $O$.

Reduction procedure

1. choose the reduction point $O$;
2. calculate the components of all forces and form the resultant vector $\vec R$;
3. calculate the moment of every force about $O$;
4. add all applied couple moments;
5. write the equivalent system $\vec R$ and $\vec M_O$;
6. if required, determine whether the system can be simplified further.

The conditions $\vec R=0$ and $\vec M_O=0$ lead directly to the general equilibrium equations for a rigid body.