Statics

Statics · Study · FE Civil · FE → PE Prep

Equilibrium is the heart of statics and the most heavily tested idea on the FE Civil exam: a body at rest has zero net force and zero net moment, full stop. Every reaction, every member force, every "is this beam balanced" question reduces to writing

\sum\mathbf{F}=0

and

\sum\mathbf{M}=0

and solving. The skill that separates fast solvers from slow ones is the free-body diagram — isolate the right body, show every force once, and the equations almost write themselves. The FE Reference Handbook's Statics chapter gives the equilibrium requirements, the truss methods, and the zero-force-member rules; this concept turns those bare statements into a working procedure.

The free-body diagram

A free-body diagram (FBD) isolates a single body and shows every external force and couple acting on it: applied loads, the weight at the center of gravity, and reactions where supports were removed. Replace each support by the forces it can transmit — a roller gives one force perpendicular to its surface, a pin gives two perpendicular force components, and a fixed support gives two force components plus a couple. The FBD is not optional scratch work; an honest FBD with consistent positive directions is what makes the sign of your answer trustworthy.

\text{roller}:1\text{ reaction}\quad \text{pin}:2\text{ reactions}\quad \text{fixed}:2\text{ forces}+1\text{ couple}

Equilibrium equations

For a 2-D rigid body, equilibrium gives three independent scalar equations — two force sums and one moment sum — so you can solve for at most three unknown reactions on a determinate body. The moment equation may be taken about any point; choosing a point where two unknowns intersect drops them out and isolates the third. In 3-D there are six equations: three force sums and three moment sums.

\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M_O=0

Trusses and the method of joints

An ideal truss carries load only through two-force members — straight, pin-connected, and loaded only at the joints — so every member is in pure tension or compression along its axis. The method of joints applies

\sum F_x=0

and

\sum F_y=0

The method of sections

When you need the force in just one or two interior members, the method of sections is far faster than chaining through joints. Cut the truss with an imaginary line that passes through no more than three members of unknown force, draw the FBD of either side, and apply the three equilibrium equations to the whole section at once. The trick is the moment equation: take moments about the point where two of the cut members intersect, and the third member's force falls out directly.

Zero-force members

Spotting zero-force members before you compute saves real time. Two rules cover almost every case: (1) if exactly two non-collinear members meet at a joint that carries no external load or reaction, both members are zero-force; and (2) if three members meet at an unloaded joint and two of them are collinear, the odd (non-collinear) member is zero-force. These members are not useless — they brace against buckling and handle alternate load cases — but for the given loading they carry nothing, and the exam loves to ask you to identify them by inspection.

\text{2 members, no load}\Rightarrow \text{both }=0;\quad \text{3 members, 2 collinear, no load}\Rightarrow \text{3rd}=0

Frames and machines

Frames and machines contain at least one multi-force member — a member loaded at more than two points, so its internal force is not a simple axial pull — which means the truss shortcut does not apply. The strategy is to dismember: draw an FBD of the whole structure to find the external reactions, then separate it into individual members and apply equilibrium to each, invoking Newton's third law so that the force one member exerts on a pin is equal and opposite to the force the pin exerts back. Machines differ from frames only in that they transmit and modify forces (think pliers or a bolt cutter), but the analysis is identical.

\mathbf{F}_{AB}=-\mathbf{F}_{BA}\quad(\text{Newton's third law at every connection})

Exam strategy

Always start with an FBD and a sign convention — tension-positive for members, CCW-positive for moments. For reactions, take moments about a support pin to kill two unknowns at once and solve a single equation for one reaction. For trusses, scan for zero-force members first, then pick the tool: method of joints when you need many member forces or the whole truss, method of sections when you need one or two interior members fast (cut through at most three unknowns, take moments at the intersection of two of them). For frames, get external reactions first, then dismember. If a member crosses your section as a slope, resolve it into horizontal and vertical components using the member's own geometry (run, rise, length).

A. Resultants of force systems

Resultants and Equivalent Force Systems

C. Equilibrium of rigid bodies

Equilibrium, Trusses, and Frames

The free-body diagram

Equilibrium equations

Trusses and the method of joints

The method of sections

Zero-force members

Frames and machines

Exam strategy

Key equations

Worked examples

Common pitfalls

References

E. Centroid of area

Centroids and Area Moments of Inertia

G. Static friction

Static Friction