5.5 : Two Force Member

The equilibrium of a two-force body is a particular case that is often encountered in practical applications. A two-force body is a rigid body that is subjected to only two external forces. For such a body to be in equilibrium, the two forces must have the same magnitude, the same line of action, and the opposite direction.

Consider any of the shown plates subjected to two forces, F₁ and F₂, that act at points A and B, respectively. If the plate is in equilibrium, then the sum of the moments of F₁ and F₂ about any axis must be zero. Summing the moments about point A, we can show that the moment of F₂ must be zero. As a result, its line of action must pass through point A. Similarly, summing the moments about point B, we can show that the line of action of F₁ must pass through point B. This means that both forces have the same line of action—the line joining A and B.

Furthermore, it can be shown that the two forces have the same magnitude and opposite directions. This can be seen from the equations ΣF_x = 0 and ΣF_y = 0, where the forces cancel each other out.

When several forces act at points A and B, the forces acting at A can be replaced by their resultant F₁, and their resultant F₂ can replace those acting at B. A two-force body can be generally defined as a rigid body subjected to forces acting at only two points. It is important to note that, in any case, the resultants F₁ and F₂ must have the same line of action, the same magnitude, and the opposite direction to ensure the equilibrium of the body.