Free-Body Diagrams & Equilibrium

What Is It?

A free-body diagram is a representation of a structural component or system, isolated from its surroundings, showing all the external forces and moments acting on it. The free-body diagram is the fundamental tool of structural mechanics — it makes the loads visible, explicit and checkable. Equilibrium is the condition that the sum of all forces and the sum of all moments acting on the body equal zero (for a static condition) or equal the inertia terms (for a dynamic condition). Together, the free-body diagram and the equilibrium equations form the basis for defining and checking structural loads.

Why It Matters

The free-body diagram is the first and most important check in any structural analysis. If the free-body diagram does not balance, the structural model should not be trusted. A finite element model that does not recover the applied loads in its reactions is not in equilibrium and its results are unreliable. The free-body diagram is also the primary tool for defining loads at interfaces — by isolating a component and drawing the forces and moments at its boundaries, the engineer defines exactly what loads the component must carry. This is essential for passing loads between models, for checking interface loads and for independent hand verification of computational results.

IF THE FREE-BODY DIAGRAM DOES NOT BALANCE, THE STRUCTURAL MODEL SHOULD NOT BE TRUSTED. The free-body diagram is the first verification check — before any stress is examined, before any margin is computed, the loads must balance.

The Equilibrium Equations

For a body in static equilibrium, the sum of all forces and the sum of all moments must equal zero. In three dimensions, this gives six equations — three force balance and three moment balance.

Static equilibrium:

  ΣFx = 0    ΣFy = 0    ΣFz = 0

  ΣMx = 0    ΣMy = 0    ΣMz = 0

where:
  Fx, Fy, Fz = force components in x, y, z
  Mx, My, Mz = moment components about x, y, z axes

For dynamic equilibrium (D'Alembert):
  ΣF = ma    ΣM = Iα

where:
  m = mass, a = acceleration
  I = mass moment of inertia, α = angular acceleration

Isolating the Component or System

The free-body diagram is created by isolating the component or system of interest. The boundaries of the free body are chosen by the engineer — they may be the physical boundaries of a component, or they may be cuts through the structure at points of interest. At every boundary, the forces and moments that act across the boundary are shown. These include: external applied loads (forces, pressures, moments), support reactions (forces and moments at constraints), interface loads (forces and moments at connections to other components), and internal forces at cuts (shear, axial, bending, torsion). The act of isolating the body and drawing these loads makes them explicit and checkable.

External Loads, Reactions, Moments and Support Conditions

The free-body diagram shows four categories of loading. External loads are the applied forces, pressures and moments from the operating environment. Reactions are the forces and moments generated at the supports — they are not independently applied but are determined by the equilibrium equations. Moments may be applied directly (a torque, a couple) or may arise from forces acting at a distance from the reference point. Support conditions define what reactions are available — a pinned support provides two force reactions (in plane), a fixed support provides three forces and three moments (in 3D), a roller provides one force reaction. The support conditions must match the physical constraint — over-constraining a model produces artificial reactions; under-constraining produces mechanisms.

Free-body diagram — simply supported beam:

     w (distributed load)
     ↓↓↓↓↓↓↓↓↓↓↓↓↓
     │            │
     A            B
     ↑            ↑
     RA           RB

Equilibrium:
  ΣFy = 0:  RA + RB − w·L = 0
  ΣMA = 0:  RB·L − w·L·(L/2) = 0

Solving:
  RB = w·L/2    RA = w·L/2

2D and 3D Equilibrium

In two dimensions, equilibrium provides three equations: two force balance (ΣFx, ΣFy) and one moment balance (ΣMz). These three equations can solve for up to three unknown reactions. In three dimensions, equilibrium provides six equations: three force balance and three moment balance, solving for up to six unknowns. If a structure has more unknowns than equilibrium equations, it is statically indeterminate — the extra reactions cannot be found from equilibrium alone and require the structural stiffness (compatibility) to solve. If a structure has fewer unknowns than equilibrium equations, it may be unstable (a mechanism) — it cannot carry load without additional support.

Why Free-Body Diagrams Remain Essential with FEA

Finite element analysis does not eliminate the need for free-body diagrams — it makes them more important, not less. The FEA model computes reactions that should balance the applied loads; the free-body diagram is the check that this has happened. The FEA model computes interface loads between components; the free-body diagram of each component verifies that the interface loads are consistent with equilibrium. The FEA model may have modelling errors — missed loads, incorrect constraints, wrong units — that the free-body diagram reveals. An engineer who relies on FEA without performing free-body checks is accepting the model on faith, not on evidence. The free-body diagram is the independent verification that the computational model is physically correct.

Common Mistakes

COMMON MISTAKE: Trusting FEA reactions without checking them against a hand free-body diagram. The FEA reaction sum should equal the applied load sum. If it does not, there is a modelling error — a missed load, an incorrect constraint, a units error — that must be found and corrected before any stress results are used.

Verification

LOAD CHECK: Sum the applied loads and the reactions from the FEA model. Do the forces balance in all directions? Do the moments balance about all axes? If not, identify the source of the imbalance before proceeding with stress analysis.

Key Takeaways

• A free-body diagram isolates a component and shows all forces and moments acting on it
• Static equilibrium requires ΣF = 0 and ΣM = 0 (six equations in 3D)
• Reactions are determined by equilibrium, not independently applied
• FEA reactions must balance applied loads — the free-body diagram is the verification
• Free-body diagrams remain essential for independent checking of computational results