Moments

For a rigid body in equilibrium, the sum of all the forces and the sum of all the moments must be zero,

     ΣF = 0       ΣM = 0

Using rectangular coordinates, these equations can be expressed by the vector equations,

     ΣF = ΣF_xi + ΣF_yj + ΣF_zk = 0

     ΣM = ΣM_xi + ΣM_yj + ΣM_zk = 0

For the expanded equation, each coefficient of the i, j, and k unit vectors must equal zero for static equilibrium.

     ΣF_x = 0      ΣF_y = 0     ΣF_z = 0

     ΣM_x = 0     ΣM_y = 0    ΣM_z = 0

The equilibrium equation has been separated into three components corresponding to the x, y, and z axes for both the forces and moments. Since each equation is independent of the others, the equations can be used to determine up to six unknowns for a full 3D problem.

"Two-dimensional" is used to describe problems in which the forces reside in a particular plane (x-y for example), and the axes of all moments are perpendicular to the plane.

Because there is no force in the z direction, and no moments about the x- or y-axis, three of the six independent scalar equations are automatically satisfied. The remaining equations for a rigid body in 2-D equilibrium are

     ΣF_x = 0      ΣF_y = 0     ΣM_z = 0

Since there is a maximum of three independent equations for a rigid body in two-dimensional equilibrium, only three unknown forces or couples can be solved.