3.1 Statics

Why Statics Dominates FE Civil

Statics carries 8–12 of the 110 FE Civil questions and feeds directly into Mechanics of Materials, Structural, and Geotechnical problems. Every reaction, member force, and stress on the exam begins with a correct free-body diagram (FBD) and the three planar equilibrium equations.

Resultants and Equilibrium

A force system reduces to a single resultant force and a couple. In two dimensions a body is in equilibrium when:

• ΣFx = 0 — horizontal forces balance
• ΣFy = 0 — vertical forces balance
• ΣM = 0 — moments about any point balance

Three equations solve at most three unknowns. Always pick a moment center that eliminates an unknown — taking moments about a pin removes both pin reaction components, leaving one equation in one unknown.

Free-Body Diagrams

The FBD is the single most important habit for FE Civil. Isolate the body, replace every support and contact with its reaction, and show all applied loads at their correct lines of action.

Trusses: Joints and Sections

A truss is a pin-connected frame of two-force members carrying only axial tension or compression. A planar truss is statically determinate and stable when:

$m + r = 2j$

where m = members, r = reactions, j = joints. If m + r < 2j the truss is unstable; if greater, it is indeterminate.

• Method of joints — write ΣFx = 0 and ΣFy = 0 at a pin (max two unknowns). Start at a joint with only two members.
• Method of sections — cut through ≤ 3 unknown members and apply ΣM to find one force directly. Best when only a few member forces are asked.

A zero-force member carries no load: at an unloaded two-member joint with non-collinear members, both are zero; at an unloaded three-member joint with two collinear members, the third is zero.

Frames and Machines

Frames contain at least one multi-force member, so members carry shear and moment, not just axial force. Dismember the structure and analyze each part separately, applying Newton's third law (equal and opposite) at connecting pins.

Centroids and Moment of Inertia

The centroid of a composite area:

$\bar{x} = \frac{\sum A_i \bar{x}_i}{\sum A_i}, \qquad \bar{y} = \frac{\sum A_i \bar{y}_i}{\sum A_i}$

The area moment of inertia resists bending. Core handbook values:

The parallel-axis theorem shifts an axis: $I = \bar{I} + A d^2$, where d is the distance between the centroidal axis and the new axis. This is essential for built-up and I-shaped sections.

Dry Friction

For Coulomb (dry) friction, the friction force resists impending motion up to a maximum:

$F_{max} = \mu_s N$

Before slipping, F is whatever value equilibrium requires (F ≤ μₛN). Once sliding, F = μₖN with μₖ < μₛ. For a block on the verge of sliding down an incline, the angle of repose equals tan⁻¹(μₛ). Look up coefficients of friction in the Statics section of the NCEES FE Reference Handbook rather than memorizing them.