Equilibrium Conditions for Coplanar Force Systems

Equilibrium of a planar force system requires both the resultant force vector and the resultant moment of the system to be zero. For a rigid body in two dimensions, this produces three independent scalar equations used to determine unknown forces and constraint reactions.

Standard equilibrium equations

For a general planar force system, the necessary and sufficient equilibrium conditions for a rigid body are:

$$\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M_O=0.$$

The point $O$ used for the moment equation may be chosen arbitrarily. All force components and moments must be entered with the adopted signs.

The first two equations eliminate the translational effect of the system along the coordinate axes, while the third eliminates its rotational effect. When all three conditions are satisfied, the body has no tendency toward translation or rotation under the forces considered.

Choosing coordinate axes

The $x$ and $y$ axes may be chosen freely, but a convenient orientation can greatly simplify the calculations. One axis is often aligned with an inclined surface, a member, or the direction of several known forces. Once the axes are selected, component signs must be used consistently.

Choosing the moment point

It is usually advantageous to choose point $O$ where the lines of action of one or more unknown reactions intersect. Their moments about that point are then zero, reducing the number of unknowns in the moment equation.

Equivalent forms

For a general planar force system, other independent sets of three equilibrium equations may be used instead of two component equations and one moment equation. For example, two moment equations about different points and one force-component equation can be valid when the selected equations remain independent. The important requirement is independence, not merely writing three equations.

Special force systems

For a concurrent planar force system, all lines of action intersect at one point, so equilibrium is described by the two independent conditions $\sum F_x=0$ and $\sum F_y=0$. For a parallel force system, an equation for force components along the common force direction together with one independent moment equation is generally sufficient.

Example

A body is subjected to a horizontal force of 8 kN to the right and an unknown force $P$ to the left, together with vertical forces of 5 kN upward and 5 kN downward. From $\sum F_x=0$, $8-P=0$, giving $P=8$ kN. The vertical condition is already satisfied. Complete equilibrium still requires $\sum M_O=0$ because a zero vector sum of forces can still leave a nonzero couple moment.

Solution procedure

1. isolate the body or system of bodies being analyzed;
2. show all applied forces and constraint reactions;
3. choose coordinate axes and a moment sign convention;
4. write the force components;
5. choose a convenient point for the moment equation;
6. form independent equilibrium equations and solve for the unknowns;
7. check signs, units and the physical meaning of the result.

Common mistakes

• omitting a support reaction;
• confusing the sign of a force with the sign of its component;
• using an incorrect moment arm;
• omitting applied couple moments;
• writing dependent equations instead of independent ones.