Force Systems and Equilibrium

FE Exam Weight: Statics accounts for 9-14 questions (~10% of the exam). Statics is fundamental to Dynamics, Mechanics of Materials, and Structural Analysis.

Vectors and Force Components

A force vector F can be resolved into rectangular components: $F_x = F \cos \theta \qquad F_y = F \sin \theta$

Magnitude and direction from components: $F = \sqrt{F_x^2 + F_y^2} \qquad \theta = \arctan\frac{F_y}{F_x}$

3D Force Components

$F_x = F \cos \alpha \qquad F_y = F \cos \beta \qquad F_z = F \cos \gamma$

where α, β, γ are the angles between the force and the x, y, z axes respectively.

Direction cosines: cos²α + cos²β + cos²γ = 1

Moment of a Force

The moment of a force about a point is the tendency of the force to cause rotation:

$M_O = F \times d$

where d is the perpendicular distance from the point to the line of action of the force.

Vector cross product form: $\vec{M}_O = \vec{r} \times \vec{F}$

where r is the position vector from O to any point on the line of action of F.

Sign Convention: Counterclockwise (CCW) is typically positive; clockwise (CW) is negative.

Varignon's Theorem

The moment of a force about a point equals the sum of the moments of its components about the same point: $M_O = F_x \cdot d_y + F_y \cdot d_x$

This is extremely useful — instead of finding the perpendicular distance, resolve the force into components and sum moments of each component.

Couples

A couple consists of two equal, opposite, parallel forces separated by distance d: $M = F \times d$

Properties of couples:

• A couple produces a pure moment (rotation only, no translation)
• The moment of a couple is the same about ANY point
• A couple can be moved anywhere on the body without changing its effect
• Couples can only be balanced by other couples

Equilibrium Conditions

For a body in static equilibrium:

2D Equilibrium: $\sum F_x = 0 \qquad \sum F_y = 0 \qquad \sum M_O = 0$

These three equations can solve for up to three unknowns in 2D.

3D Equilibrium: $\sum F_x = 0 \quad \sum F_y = 0 \quad \sum F_z = 0$ $\sum M_x = 0 \quad \sum M_y = 0 \quad \sum M_z = 0$

Six equations → up to six unknowns in 3D.

Free-Body Diagrams (FBDs)

Steps to draw an FBD:

1. Isolate the body (or portion of a body)
2. Show all external forces (applied loads, weights)
3. Replace supports with their reaction forces
4. Include dimensions and angles
5. Label all forces and moments

Common Support Reactions

Exam Strategy: When solving for unknown reactions, choose your moment point wisely. Taking moments about a point where two unknowns intersect eliminates both from the equation, giving you the third unknown directly.