Equilibrium

A body in equilibrium has no net force and no net moment. Learn free-body diagrams, support types and their reactions, solving beam and bracket reactions, and shear-force and bending-moment diagrams.

Most structures and machine parts are designed to stay still — or move steadily — under load. That is equilibrium: all forces and all moments cancel. Writing and solving the equilibrium equations tells you the reaction forces at supports and the internal forces inside members.

The single most important skill in mechanics is drawing a correct free-body diagram (FBD): isolate the body, show every external force and support reaction, then apply ΣF = 0 and ΣM = 0.

An analogy

A tug-of-war that isn't moving: both teams pull hard, but the forces balance, so the rope stays put. Equilibrium doesn't mean no forces — it means balanced forces.

Free-body diagrams

Draw the body alone. Replace every support and contact with the forces it can exert. Add all loads, including weight. Choose axes and a sign convention. Only then write equations.

Supports and reactions

Each support type prevents certain motions and provides matching reactions.

• Roller: one reaction, perpendicular to the surface.
• Pin/hinge: two force components (x and y).
• Fixed (built-in): two force components plus a moment.

Solving strategy

Take moments about a point where unknowns meet — they vanish from that equation. Use the remaining force equations to find the rest, and check with an extra moment equation. A 2D rigid body gives three independent equations, so three unknown reactions can be found (statically determinate).

Shear force and bending moment diagrams

Cutting a beam at x and balancing one side gives the internal shear V(x) and bending moment M(x). The maximum bending moment (often where V crosses zero) sizes the beam in strength-of-materials design.

Statically indeterminate structures

With more unknowns than equations (a beam on three supports, a propped cantilever), equilibrium alone isn't enough; deformation compatibility is added (force/flexibility methods, slope-deflection, stiffness/FEM). Indeterminate structures are stiffer and more redundant, which is why most real buildings are indeterminate.

Formulas

• 2D equilibrium: ΣF_x = 0, ΣF_y = 0, ΣM = 0
• Simply supported beam, point load P at a: R_A = P(L − a)/L, R_B = P·a/L
• UDL w over span L: R_A = R_B = wL/2, M_max = wL²/8
• Shear–moment relation: dM/dx = V, dV/dx = −w

Worked example: Beam reactions

Given: Span L = 6 m; Point load 12 kN at 2 m from A

Find: R_A, R_B and the maximum bending moment.

• ΣM_A = 0: R_B × 6 = 12 × 2 → R_B = 4 kN
• ΣF_y = 0: R_A = 12 − 4 = 8 kN
• M_max at load: 8 × 2 = 16 kN·m

Answer: R_A = 8 kN, R_B = 4 kN, M_max = 16 kN·m

Worked example: Uniformly loaded beam

Given: w = 5 kN/m over L = 6 m

Find: Reactions and maximum moment.

• Total load = 30 kN → R_A = R_B = 15 kN
• M_max = wL²/8 = 5 × 36/8 = 22.5 kN·m at mid-span

Answer: 15 kN each; 22.5 kN·m

Common mistakes

• Wrong: Forgetting a reaction or the body's own weight in the FBD. — Right: Replace every support with all the reactions it can provide, and include self-weight when significant.
• Wrong: Placing a distributed load's resultant at the wrong point. — Right: A UDL's resultant (w × length) acts at the middle of the loaded length.

Applications

• Buildings & bridges: Every beam, column and foundation is sized from reactions and bending moments found by equilibrium.
• Machine frames: Bearing loads on shafts are support reactions from gear and pulley forces.
• Lifting equipment: Outrigger and wheel reactions on cranes decide whether the ground and the machine can carry the lift.

Key takeaways

• Equilibrium: ΣF = 0 and ΣM = 0.
• A correct FBD is the whole battle.
• Rollers 1, pins 2, fixed supports 3 reactions.
• V and M diagrams locate the critical section for design.

Going further

• Virtual work: Equilibrium can also be expressed as zero virtual work for any admissible virtual displacement — elegant for mechanisms and the foundation of FEM.
• Trusses: Method of joints (ΣF = 0 at each pin) and method of sections (cut and take moments) find member forces in bridges and roof trusses.
• Influence lines: For moving loads (vehicles on a bridge), influence lines show how a reaction or moment varies as a unit load travels across, giving worst-case positions.