Free body diagram; Varignon's theorem

Free body diagram

A free body diagram (FBD) is a sketch of a body, or of part of a structure, isolated from everything around it, with all external forces and moments acting on it shown: applied loads, weight, support reactions and contact forces. It is the first step in solving any statics problem.

Sketch 1: simply supported beam with a load P

Real beam:            FBD of beam:
     P                     P
     |                     |
     v                     v
 A o-----o B           ===========
 /\    (roller)         ^         ^
                       A_y       B_y
                  (A_x to the right at A)

Sketch 2: block on a rough incline. The FBD shows weight W down, normal reaction N perpendicular to the plane and friction F along the plane up the slope.

Varignon's theorem

Statement: the moment of a force about any point equals the algebraic sum of the moments of its components about the same point. For a system of concurrent forces, the moment of the resultant about a point equals the sum of the moments of the individual forces about that point.

Proof (two concurrent forces): Let forces P and Q act at point A, with position vector r from the moment centre O. Their resultant is R = P + Q.

Moment of R about O:   M_O = r × R
                           = r × (P + Q)
                           = r × P + r × Q      (distributive law of the cross product)

So the moment of the resultant equals the sum of the moments of P and Q about O. Applying it repeatedly extends it to any number of forces: r × ΣF = Σ(r × F).

Proof in 2D by components: let a force F at (x, y) have components Fx and Fy. Its moment about the origin is M_O = x·Fy − y·Fx. This is the sum of the moments of Fy (= x·Fy, anticlockwise) and Fx (= −y·Fx), which proves the theorem.

Use: the moment of an inclined force can be found easily by resolving it into horizontal and vertical components, and the line of action of a resultant can be located.