Free-Body Diagrams and Equilibrium

Statics is the study of bodies at rest. Almost every mechanics question in GATE ME starts the same way: isolate the body, draw its free-body diagram (FBD), and write the equilibrium equations.

The free-body diagram

An FBD shows one body, cut free from everything it touches, with every force acting on it:

1. Its weight, at the centre of gravity.
2. Applied loads.
3. A reaction at every support or contact that was removed.

Equations of equilibrium

A body in a plane is in equilibrium when

ΣFx = 0, ΣFy = 0, ΣM = 0 (about any point).

That gives at most three independent equations per body, so three unknowns can be found. Take moments about a point where two unknown forces meet: each moment equation then has only one unknown.

Special members

• Two-force member: forces act at only two points (for example a pin-ended link with no load in between). The two forces are equal, opposite and along the line joining the points.
• Three-force member: three non-parallel forces in equilibrium must be concurrent (pass through one point).
• Lami's theorem: for three concurrent forces in equilibrium, each force is proportional to the sine of the angle between the other two: P/sin α = Q/sin β = R/sin γ.

Worked example

A 100 N lamp hangs from two strings making 30° and 60° with the ceiling. Find the tensions.

Resolve forces at the knot, with T₁ the string at 30° and T₂ the string at 60°:

• Horizontal: T₁ cos 30° = T₂ cos 60°. Vertical: T₁ sin 30° + T₂ sin 60° = 100.
• From the first, T₂ = √3 T₁. Then T₁/2 + (√3 T₁)(√3/2) = 100, so 2T₁ = 100: T₁ = 50 N, T₂ = 86.6 N.

The string closer to vertical carries more load, which is a quick sanity check.

Simply supported beam reactions

A beam of span L with a point load P at distance a from the left support:

• Moments about the left support: R_B × L = P × a, so R_B = Pa/L.
• Vertical forces: R_A = P − R_B = P(L − a)/L.

Practice