FBD and equilibrium: meaning and importance

Free body diagram (FBD)

A free body diagram is a sketch of a body, or part of a structure, isolated from its surroundings, on which all external forces and moments are shown: applied loads, self-weight, and the reactions of the removed supports and neighbouring members. Typical reactions: roller gives one force normal to the surface, hinge gives two components, fixed support gives two components and a moment, cable gives a tension.

Beam with load            FBD of the beam
   P                           P
   |                           |
   v                           v
 A o=========o B          ===========
 /\        (roller)        ^        ^
                           A_y      B_y   (and A_x at A)

Physical meaning of equilibrium

A body is in equilibrium when the resultant force and the resultant moment acting on it are zero (ΣF = 0, ΣM = 0). Physically, the body has no tendency to start or change translation or rotation. It remains at rest (static equilibrium) or moves with constant velocity. All actions on the body are balanced by reactions.

Importance in structural analysis

Of the free body diagram

• It shows all forces on a part of a structure clearly, so no force is missed or repeated.
• It gives the unknown reactions and internal forces in the correct sense.
• Cutting a member and drawing the FBD of one side gives axial force, shear force and bending moment at that section.
• It is the first and most important step in any statics problem.

Of equilibrium

• The equations ΣFx = 0, ΣFy = 0, ΣM = 0 give the support reactions of beams, frames and trusses.
• They give member forces in trusses (method of joints, method of sections).
• A structure that cannot be in equilibrium under its loads is unstable and would collapse.
• Counting the unknowns against the equations decides whether a structure is statically determinate.
• Overturning and sliding of walls, dams and foundations are checked using equilibrium.