Statics: Free-Body Diagrams and Equilibrium

Build the free-body diagram first

FE Mechanical statics questions often look short because the arithmetic is simple after the model is right. Start by isolating one body, removing supports, and replacing each contact with the reaction components it can actually provide. A smooth roller gives one normal reaction, a pin gives two force components, and a fixed support gives two force components plus a couple moment in a 2D model.

Declare axes and signs before using the calculator. A common convention is +x to the right, +y upward, and counterclockwise moments positive. The convention is not sacred, but it must be consistent from the diagram through the final answer. If your support reaction comes out negative, that usually means the actual direction is opposite your assumed arrow, not that equilibrium failed.

Equilibrium equation sequence

For a rigid body in a plane, use three independent equations: sum F_x = 0, sum F_y = 0, and sum M = 0. Choose the moment center strategically. Taking moments about a pin support removes both pin reactions from the equation, often leaving one unknown.

For distributed loads, do not put the load into force equilibrium as w until it has been converted to a resultant. A rectangular load has resultant wL at its midpoint. A triangular load has resultant one-half w_max L located one-third of the base length from the larger end. Keep the original distributed load on the sketch until you have recorded both magnitude and location.

FE handbook and calculator workflow

The FE Reference Handbook is useful for area centroids, moments of inertia, and standard beam relationships, but basic equilibrium is usually not a lookup problem. Use the handbook for geometry properties and standard resultants; use your free-body diagram to decide which equation applies.

Units are a frequent trap. Moments must be force times distance: N-m, lb-ft, or lb-in. If a beam length is in meters and a load is in newtons, do not convert only one distance to millimeters because a section property table happens to use mm. In USCS dynamics, distinguish lbm from lbf; in statics, weights are forces, so a 200 lb load normally means 200 lbf unless the question states mass.

A repeatable statics pass

Use this short routine on every equilibrium problem:

1. Isolate the body and draw all external forces, couples, dimensions, and angles.
2. Replace supports and distributed loads with the correct reaction model or resultant.
3. Pick axes and a positive moment direction.
4. Write symbolic equilibrium equations before substituting numbers.
5. Solve, then check signs, units, and whether the result is physically plausible.

A plausible answer balances both force and moment. If the vertical reactions add to the total downward load but the moment equation is not satisfied, the reactions are still wrong. If the reaction is larger than the total load, ask whether a large overturning moment or long lever arm explains it before changing signs blindly.