Addition of Forces. Resultant of a Force System

Forces are added according to the rules of vector addition. The result of this vector addition is the resultant vector of the force system. For a concurrent force system, whose lines of action pass through a common point, this vector can be applied at the point of concurrency and is the resultant force of the system.

Geometric addition

Two forces can be added using the parallelogram or triangle rule. For several forces, construct a force polygon by placing each successive vector at the end of the previous one. The vector from the start of the first force to the end of the last is their geometric sum.

Analytical addition in two dimensions

In Cartesian coordinates, first sum the components algebraically:

$$R_x=\sum_i F_{ix},\qquad R_y=\sum_i F_{iy}.$$

For a planar force system, after algebraically summing the components $R_x=\sum F_{ix}$ and $R_y=\sum F_{iy}$, the magnitude of the resultant vector is obtained from the Pythagorean relation:

$$R=\sqrt{R_x^2+R_y^2}$$

This formula gives the magnitude of the vector sum of the forces. For a general force system, its moment effect must also be considered separately.

The direction of the resultant vector is determined from its components while accounting for the correct quadrant, for example with $\operatorname{atan2}(R_y,R_x)$.

Resultant force and resultant vector

The vector sum of all forces always defines the resultant vector $\vec R=\sum\vec F_i$. For a general force system, however, this vector alone is not sufficient to replace the entire system by one force because the moment effect must also be preserved. The term resultant force should therefore be used when a single force is actually equivalent to the original system.

Collinear forces

For forces acting along one line, choose a positive direction and add the forces algebraically. For example, forces of 8 kN and 5 kN acting in the same direction give 13 kN. If they act in opposite directions, the magnitude of the sum is 3 kN and its direction is that of the larger force.

Example with perpendicular forces

Suppose forces of 3 kN along $+x$ and 4 kN along $+y$ act at a common point. Then $R_x=3$ kN and $R_y=4$ kN, so the resultant magnitude of the concurrent system is $R=\sqrt{3^2+4^2}=5$ kN. Its direction is $\alpha=\arctan(4/3)\approx53.1^\circ$ from the positive $x$ axis.

Common mistakes

• adding magnitudes of non-collinear forces as ordinary numbers;
• ignoring the signs of force components;
• using only $\arctan(R_y/R_x)$ without checking the quadrant;
• treating the resultant vector of a general force system as a single equivalent force without checking moments.

The next step in statics is reduction of a general force system to a specified point, where the resultant moment is considered together with the resultant vector.