Force Systems, Free-Body Diagrams & Equilibrium Notes

Last reviewed 16 Sept 2026 · 7 min read

Mechanics, force and rigid bodies

Engineering mechanics studies forces and their effects on bodies. Statics deals with bodies at rest (or moving uniformly); dynamics with accelerated motion. In statics we usually treat bodies as rigid — the distance between any two points does not change — which is accurate enough for finding reactions and equilibrium.

A force is an action that changes, or tends to change, the state of rest or motion of a body. It is a vector defined by four characteristics:

Magnitude (N, kN)
Direction (line of action and angle)
Sense (which way along the line)
Point of application

DefinitionPrinciple of transmissibility

The external effect of a force on a rigid body is unchanged if the force is moved anywhere along its line of action. (It is not true for internal stresses or for deformable bodies.)

Classification of force systems

System	Description	Example
Collinear	All forces on one line	Forces in a rope in tug of war
Coplanar concurrent	In one plane, lines meet at a point	Forces at a truss joint
Coplanar parallel	In one plane, parallel lines	Loads and reactions on a beam
Coplanar non-concurrent, non-parallel	General plane system	Ladder against a wall
Non-coplanar (spatial)	In three dimensions	Forces on a tripod or tower

Resultant of forces

The resultant is the single force that produces the same external effect as the whole system.

Parallelogram and triangle laws

Two forces $P$ and $Q$ at angle $θ$ :

R = P^{2} + Q^{2} + 2 P Q cos θ, tan α = \frac{Q sin θ}{P + Q cos θ}

where $α$ is the angle of $R$ from $P$ .

$θ = 0°$ : $R = P + Q$ (maximum). $θ = 180°$ : $R = ∣ P - Q ∣$ (minimum). $θ = 90°$ : $R = P^{2} + Q^{2}$ .
If $P = Q$ : $R = 2 P cos (θ /2)$ along the bisector.

The triangle law: if two forces are drawn tip to tail, the closing side (from the start of the first to the end of the second) is the resultant. The polygon law extends this to many forces.

Resolution of forces

A force $F$ at angle $θ$ to the $x$ -axis has components $F_{x} = F cos θ$ , $F_{y} = F sin θ$ . For a coplanar concurrent system:

R = (\sum F_{x})^{2} + (\sum F_{y})^{2}, tan α = \frac{\sum F _{y}}{\sum F _{x}}

Watch the signs of $\sum F_{x}$ and $\sum F_{y}$ to place $α$ in the correct quadrant.

Moment of a force and couples

The moment of a force about a point is its turning effect: $M = F \times d$ , where $d$ is the perpendicular distance from the point to the line of action. Units N·m. Clockwise and anticlockwise moments are given opposite signs.

FormulaVarignon's theorem (principle of moments)

The moment of the resultant of a system of forces about any point equals the algebraic sum of the moments of the individual forces about the same point.

R \cdot \overset{ˉ}{d} = \sum F_{i} d_{i}

Used to locate the line of action of the resultant of parallel or general forces.

A couple is two equal, opposite, parallel, non-collinear forces. Its moment $= F \times a$ ( $a$ = arm) and is the same about every point in the plane. A couple produces rotation only; it cannot be balanced by a single force — only by another couple.

Resolution of a force into a force and a couple: a force $F$ at A is equivalent to an equal force at B plus a couple $F \times d$ ( $d$ = perpendicular distance between the lines). This is how an eccentric load on a column is replaced by an axial load plus a moment.

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Mechanics, force and rigid bodies

Classification of force systems

Resultant of forces

Parallelogram and triangle laws

Resolution of forces

Moment of a force and couples

This chapter is in the syllabus of