Equilibrium Conditions for Three-Dimensional Force Systems

Equilibrium of a rigid body in three dimensions requires the simultaneous absence of translational and rotational effects of the force system. In vector form, this means $\sum\vec F=0$ and $\sum\vec M_O=0$. Projection onto the three coordinate axes gives six scalar conditions.

General equilibrium equations

For a general three-dimensional force system, rigid-body equilibrium is described by six scalar equations:

$$\sum F_x=0,\quad \sum F_y=0,\quad \sum F_z=0,$$

$$\sum M_x=0,\quad \sum M_y=0,\quad \sum M_z=0.$$

The first three equations state that the resultant force vector is zero, while the last three state that the resultant moment about the selected reference point is zero.

For a single rigid body, these six independent equations are the basic tool for determining unknown reactions of three-dimensional supports and unknown loads.

Force components

Before writing the equations, each three-dimensional force is expressed through its components $F_x$, $F_y$, and $F_z$. If a force direction is specified by two points, first determine the unit vector along its line of action and then multiply it by the force magnitude.

Moment components

The moment of a force about point $O$ is calculated as $\vec M_O=\vec r\times\vec F$. The resulting moment vector is then resolved into $M_x$, $M_y$, and $M_z$. Applied couple moments are added directly to the corresponding components of the total moment.

Choosing the moment reference point

As in planar statics, the moment reference point can be selected for convenience. If the lines of action of several unknown forces pass through the chosen point, their moments about that point vanish. In three dimensions, however, all three components of the moment vector must be handled consistently.

Reactions of three-dimensional constraints

The number of unknown reactions depends on the motions prevented by the constraint. For example, an idealized fixed support in three dimensions can transmit three force components and three moment components. The specific support model must be established before equilibrium equations are written.

Special force systems

For a concurrent three-dimensional force system, all force lines of action pass through one point, so the three force-component equations are the independent equilibrium conditions. Other special geometries may reduce the number of required equations, but the general set of six equations remains the fundamental form.

Example

Suppose three forces acting at a point have components $\vec F_1=(3,0,-4)$ kN, $\vec F_2=(-3,5,0)$ kN, and $\vec F_3=(0,-5,4)$ kN. The sums of all three components are zero, so this concurrent force system is balanced. For a general nonconcurrent system, the three moment equations must also be checked.

Solution procedure

1. construct a three-dimensional free-body diagram;
2. replace all constraints by the appropriate reactions;
3. express forces in Cartesian components;
4. choose a reference point $O$ and calculate force moments;
5. write $\sum F_x=\sum F_y=\sum F_z=0$;
6. write $\sum M_x=\sum M_y=\sum M_z=0$;
7. solve the equations and check the physical meaning of the calculated reactions.

Common mistakes

• omitting one of the three force or moment components;
• determining the force unit vector in the wrong direction;
• confusing a moment about a point with a moment about an axis;
• reversing the order of the cross product $\vec r\times\vec F$;
• using six equations without first modeling the three-dimensional constraints correctly.