FREE FE Mechanical Study Guide 2026: All 14 Areas

This free FE Mechanical study guide teaches the breadth of mechanical engineering the NCEES Fundamentals of Engineering (FE) exam tests, organized to the current NCEES FE Mechanical exam specifications and their 14 knowledge areas.^[1] The FE Mechanical exam is typically the first step toward Professional Engineer (PE) licensure.

It’s interactive, not a wall of text: every module has built-in checkpoint quizzes, flashcards, and practice questions, so you review by doing — not just reading. The single most important habit is to study with the FE Reference Handbook open, because it is the only reference you get on exam day.

What the FE Mechanical Exam Is

The FE Mechanical exam is a computer-based test (CBT) with 110 questions and a 6-hour appointment (a short agreement, an 8-minute tutorial, an optional 25-minute break, and 5 hours 20 minutes of testing).^[2] It is closed book, but you get a searchable electronic copy of the FE Reference Handbook — the only reference allowed.^[3] It also uses both SI and US Customary units, so consistent unit conversion is a tested skill.

The most useful thing to know before you study: the FE rewards breadth and speed, not depth. Most items are single-concept problems you can solve in a couple of minutes if you can find the right equation in the Handbook quickly. Practicing that lookup-and-solve loop is what turns knowledge into a passing score.

FE Mechanical Exam Snapshot

The NCEES specifications group the exam into 14 knowledge areas.^[1] Study the breadth, but weight your time toward the heavy mechanical-engineering areas — the six largest together are roughly two-thirds of the exam:

1 · Mathematics

6–9 questions. Mathematics is the toolbox under every other area: calculus, differential equations, linear algebra, and numerical methods. The questions are usually direct, and the Handbook supplies the formulas, so this is high-value review.

1.1 Calculus & Differential Equations

Expect FE items on derivatives, integrals, and the solution of ordinary differential equations (ODEs). A core fact: the number of arbitrary constants in an ODE’s general solution equals the orderof the equation — one for a first-order ODE, two for a second-order. Partial derivatives treat the other variables as constants, and the gradient of a scalar field points in the direction of steepest increase.

1.2 Linear Algebra & Numerical Methods

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ the determinant is $ad - bc$, and a nonzero determinant means the matrix is invertible. Eigenvalues solve $\det(A - \lambda I) = 0$. In numerical methods, Simpson’s one-third rule fits a parabola through three points (and needs an even number of subintervals), while the trapezoidal rule fits straight lines and is less accurate; Newton’s method finds roots iteratively and converges quadratically near a simple root.

2 · Probability & Statistics

4–6 questions. Engineering statistics on the FE centers on describing data, quantifying uncertainty, and fitting trends — the everyday tools of quality and reliability work.

2.1 Distributions & Central Tendency

Know the measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation), and the empirical 68–95–99.7 rule for the normal distribution: about 68% of values lie within one standard deviation of the mean, 95% within two, 99.7% within three. The z-score— that is, the standardized value $z = (x - \mu)/\sigma$ — counts how many standard deviations an observation sits from the mean.

2.2 Confidence Intervals & Regression

A confidence intervalis a statement about a method: a 95% interval is produced by a procedure that captures the true mean 95% of the time over repeated sampling — not a claim about one future part. Increasing the sample size narrows the interval (the margin of error scales as $\sigma/\sqrt{n}$). In regression, the coefficient of determination $R^2$ is the fraction of the variation in y explained by the model.

3 · Ethics & Professional Practice

4–6 questions. Ethics items are quick points if you know the canons. They test the NCEES Model Law and Model Rules, intellectual property, and societal considerations.^[4]

3.1 Public Safety & the Canons

The first and overriding rule: engineers must hold paramount the safety, health, and welfare of the public. This duty outranks obligations to a client or employer, so when a cost-saving change would create a serious hazard, the engineer must notify the client and the appropriate authority and refuse to proceed unsafely — documenting and proceeding does not discharge the duty.

3.2 Competence, Conflicts & Sealing

Engineers perform services only in their areas of competence, act as faithful agents (avoiding and disclosing conflicts of interest), and issue objective and truthful public statements. A licensee may seal only work prepared by or under their responsible charge — sealing unreviewed outside work is a prohibited misrepresentation.

4 · Engineering Economics

4–6 questions. Engineering economics compares alternatives by making cash flows equivalent in time. A handful of formulas and concepts cover most items.

4.1 Time Value of Money

The core idea is the time value of money: a sum has different worth at different times, so cash flows must be discounted to a common point before comparison ($P = F/(1+i)^n$). The rate of return is the interest rate that makes the present worth of all cash flows zero, and a public project is generally justified when its benefit-cost ratio is ≥ 1.0.

4.2 Alternatives, Cost Types & Inflation

To compare mutually exclusive alternatives, convert each to an equivalent uniform annual cost and pick the lowest. Distinguish a sunk cost (already spent, irrelevant to future decisions) from an opportunity cost (the value of the best forgone alternative, which does matter). And remember inflation reduces the real purchasing power of a fixed future sum.

5 · Electricity & Magnetism

5–8 questions. Mechanical engineers still need circuit and field fundamentals — the FE keeps these at an introductory level.

5.1 DC Circuits — Ohm & Kirchhoff

Ohm’s law ($V = IR$) and the two Kirchhoff laws drive DC circuit analysis: KCL says the currents entering a node equal those leaving (charge conservation), and KVL says the voltages around a loop sum to zero (energy conservation). Resistors add in series and combine as reciprocals in parallel; power is $P = VI = I^2 R$.

5.2 Fields, Induction & AC

Coulomb’s law makes the force between point charges fall off with the square of the distance. Magnetic flux density (B) is flux per unit area, and Faraday’s law says induced EMF is proportional to the rate of changeof magnetic flux — a steady flux induces nothing. In DC steady state a capacitor is an open circuit and an inductor is a short.

6 · Statics

9–14 questions — a heavy area. Statics is the study of forces in equilibrium, and it underpins mechanics of materials and machine design.

6.1 Equilibrium, Couples & Members

A rigid body in planar equilibrium gives three independent equations ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$), enough for three unknown reactions. A couple is two equal, opposite, parallel forces producing a pure moment, and a two-force member can carry force only along the line joining its two load points.

6.2 Trusses, Centroids & Friction

Solve trusses with the method of joints (two force equations per pin joint) or the method of sections (cut through and use equilibrium to find a specific interior member). Know how to find a centroid and moment of inertia (including the parallel-axis theorem), and that the maximum static friction before sliding is $F_{max} = \mu_s N$.

7 · Dynamics, Kinematics & Vibrations

10–15 questions — the largest area, tied with fluids, thermo, and design. Dynamics adds motion to the force picture: how bodies move and what forces produce that motion.

7.1 Kinematics & Newton’s Second Law

Kinematics describes motion (position, velocity, acceleration) without reference to its cause; Newton’s second law ($\sum F = ma$) then connects the net force to the resulting acceleration. In curvilinear motion, tangential acceleration changes speed while normal (centripetal) acceleration $a_n = v^2/\rho$ changes direction.

7.2 Energy, Momentum & Vibrations

Two powerful shortcuts avoid solving the full equations of motion: the work-energy theorem (net work = change in kinetic energy) and the impulse-momentum principle. In a perfectly elastic collision both momentum and kinetic energy are conserved; the coefficient of restitution measures how much relative speed survives. For vibrations, resonance occurs when a forcing frequency approaches the natural frequency.

8 · Mechanics of Materials

9–14 questions — a heavy area. Mechanics of materials finds the internal stress and strain that loads create inside a part.

8.1 Stress, Strain & Beam Diagrams

Axial load gives $\sigma = P/A$; Hooke’s law relates stress and strain in the elastic region ($\sigma = E\varepsilon$). On a beam, the relationships $dV/dx = -w$ and $dM/dx = V$ govern the shear and moment diagrams: a uniform load makes shear vary linearly, and the bending moment peaks where the shear crosses zero. The flexure formula $\sigma = Mc/I$ gives bending stress, maximum at the outer fiber.

8.2 Transformations & Column Buckling

Mohr’s circle graphically gives the principal stresses (on planes where shear is zero) and the maximum in-plane shear stress. For slender columns, the Euler buckling load $P_{cr}= \pi^2 EI/(KL)^2$ shows buckling is highly sensitive to length — doubling the length quarters the load — with the effective length factor K set by the end conditions (2.0 for fixed-free, 1.0 for pinned-pinned).

9 · Material Properties & Processing

7–11 questions. This area links the numbers from a tensile test to the microstructure and heat treatments that produce them.

9.1 The Stress-Strain Curve & Properties

Read the engineering stress-strain curve cold: the elastic slope is the modulus of elasticity, the yield strength is found by the 0.2% offset method, the peak is the ultimate tensile strength, and fracture follows necking. Percent elongation measures ductility.

9.2 Iron-Carbon Diagram & Heat Treating

On the iron-carbon diagram, alloys below about 2.1% carbon are steels and above are cast irons, with 0.76% the eutectoid composition. Slow cooling of eutectoid austenite yields lamellar pearlite; rapid quenching traps carbon to form hard, brittle martensite, which tempering then softens for toughness.

10 · Fluid Mechanics

10–15 questions — a top-weight area. Fluid mechanics spans fluids at rest and in motion, internal and external flow, and compressible flow.

10.1 Fluid Statics & Buoyancy

Hydrostatic pressure rises with depth, $P = \rho g h$, and acts equally in all directions at a point. The buoyant force (Archimedes) equals the weight of displaced fluid, so a floating body’s buoyant force exactly equals its weight.

10.2 Flow, Bernoulli & Compressible Flow

The Reynolds number ($Re = \rho V D/\mu$) predicts laminar vs. turbulent flow, and laminar pipe flow has a parabolic velocity profile (max at the centerline, twice the average). Bernoulli’s equation with continuity ($A_1 V_1 = A_2 V_2$) solves most internal-flow problems. In compressible flow, the Mach number sets the regime, and across a normal shock the Mach number drops to subsonic while pressure, temperature, and entropy rise.

11 · Thermodynamics

10–15 questions — a top-weight area. Thermodynamics covers energy, the laws, property states, and the power and refrigeration cycles built on them.

11.1 Laws, Properties & States

The first law conserves energy ($\Delta U = Q - W$); the second law sets direction — heat flows hot-to-cold, no cyclic engine reaches 100% efficiency, and entropy rises in real processes. Read steam-table states: a compressed (subcooled) liquid sits below the saturation temperature at its pressure, a superheated vapor above it.

11.2 Power & Refrigeration Cycles

Recognize the cycles by their four-process signatures: the Rankine (steam), Brayton (gas turbine), and Otto (spark ignition) cycles, with the Carnot cycle setting the reversible efficiency ceiling $\eta = 1 - T_C/T_H$. Refrigerators and heat pumps are rated by coefficient of performance, which can exceed 1.

12 · Heat Transfer

7–11 questions. Heat transfer applies thermodynamics to rates— how fast energy moves and how to enhance or resist it.

12.1 The Three Modes

Know the three modes and their governing laws: conduction (Fourier, $q = -kA\,dT/dx$), convection (Newton’s cooling, $q = hA\,\Delta T$), and radiation (Stefan-Boltzmann, $q = \varepsilon\sigma A T^4$). Radiation is the only mode that crosses a vacuum, and it scales with the fourth power of absolute temperature.

12.2 Heat Exchangers & Fins

For heat exchangers, the LMTD method needs both outlet temperatures, so when the outlets are unknown the effectiveness-NTU method gives a direct, non-iterative solution. Fins increase the area for convection and work best with a high base conductivity and a low convection coefficient (e.g., air cooling).

13 · Measurements, Instrumentation & Controls

5–8 questions. This area covers sensors, control systems, dynamic response, and the quality of a measurement.

13.1 Control Systems & PID

A PID controller sums three actions: proportional (present error), integral (accumulated past error), and derivative (rate of change). The integral action eliminates steady-state offset, while raising the proportional gain shrinks offset but makes the loop more oscillatory. A system’s transfer function is the ratio of output to input Laplace transforms with zero initial conditions.

13.2 Measurement Error & Uncertainty

Distinguish accuracy from precision: accuracy is closeness to the true value, precision is repeatability — an instrument can be precise yet biased. A systematic erroris a consistent one-directional bias (averaging won’t remove it), while a random error is unpredictable scatter (averaging reduces it).

14 · Mechanical Design & Analysis

10–15 questions — a top-weight area. Design ties the mechanics and materials areas together: choosing and sizing components so they don’t fail.

14.1 Factor of Safety & Failure Theories

The factor of safety is a material strength divided by the actual working stress — greater than 1 means a margin against failure. For ductile metals, the von Mises (distortion-energy) theory predicts yielding most accurately by combining all stress components into one equivalent stress; the maximum-normal-stress theory is the brittle-material counterpart.

14.2 Fatigue, Machine Elements & GD&T

Fatigue design uses the Goodman diagram (alternating vs. mean stress, line from the endurance limit to the ultimate strength) and accounts for stress concentrations. Know the common machine elements — springs ($k = F/\delta$), bearings, and power transmission ($P = T\omega$) — plus pressure-vessel hoop stress ($\sigma_h = pr/t$) and GD&T for interpreting drawings.

How to Use This FE Mechanical Study Guide

The FE rewards broad competency worked fast, so this guide is built to be practiced, not just read:

• Cover the breadth. Every one of the 14 areas can appear — don’t leave gaps in ethics, economics, or the math/probability areas, which are fast points.
• Lead with the heavy six. Statics, Dynamics, Mechanics of Materials, Fluid Mechanics, Thermodynamics, and Mechanical Design are roughly two-thirds of the exam.
• Study with the Handbook open. It is the only reference you get on exam day, so practice finding equations in it until lookup is automatic.
• Mind your units. The FE mixes SI and USCS — convert consistently, and check that your answer’s units come out right.
• Take every checkpoint. The end-of-module quizzes show exactly which areas need another pass, and finishing them raises your readiness score.
• Then prove it. Send weak areas into the flashcards and a full practice test, and read every rationale.

FE Mechanical Glossary

The high-yield FE Mechanical terms and formulas in one place — hover any dotted term in the guide, or flip the whole deck here as a self-grading flashcard set.

Accuracy vs. precision

Accuracy is closeness to the true value; precision is the repeatability of readings to one another.

Bernoulli's equation

Conservation of mechanical energy along a streamline for steady, incompressible, inviscid flow: $P + \tfrac{1}{2}\rho V^2 + \rho g z$ is constant.

Brayton cycle

The ideal gas-turbine cycle: two isentropic and two constant-pressure processes.

Buoyant force

The upward force on a submerged or floating body equal to the weight of fluid displaced (Archimedes' principle).

Carnot efficiency

The maximum efficiency of any engine between two reservoirs, $\eta = 1 - T_C/T_H$.

Centroid

The geometric center of an area or volume, about which the first moment of area is zero.

Coefficient of restitution

The ratio of relative separation velocity to relative approach velocity in a collision; 1 is perfectly elastic, 0 perfectly plastic.

Couple

Two equal, opposite, parallel forces that produce a pure moment with no net force; its moment is the same about every point.

Effective length factor (K)

A factor accounting for column end conditions: 1.0 pinned-pinned, 0.5 fixed-fixed, 2.0 fixed-free.

Effectiveness-NTU method

A heat-exchanger method preferred when inlet temperatures are known but outlet temperatures are not, giving a direct (non-iterative) solution.

Endurance limit

The cyclic stress amplitude below which a material can endure effectively infinite cycles without fatigue failure.

Entropy

A property measuring energy unavailability or disorder; it increases in every irreversible process.

Euler buckling load

The critical axial load for a slender column, $P_{cr} = \pi^2 EI/(KL)^2$.

Factor of safety

A material strength divided by the actual working stress; a value above 1 gives a margin against failure.

FE Reference Handbook

The only reference allowed in the FE exam — a searchable electronic document of equations and data; closed-book otherwise. Knowing where each equation lives is the core exam skill.

First law of thermodynamics

Energy conservation: for a closed system $\Delta U = Q - W$.

Flexure formula

The bending stress in a beam, $\sigma = Mc/I$, maximum at the outer fiber.

Fundamentals of Engineering (FE) exam

The NCEES computer-based exam that is typically the first step toward Professional Engineer (PE) licensure; the Mechanical discipline covers 14 knowledge areas.

GD&T

Geometric Dimensioning and Tolerancing (ASME Y14.5) — a symbolic drawing system specifying allowable feature form, orientation, and location.

Goodman diagram

A fatigue-design plot of alternating vs. mean stress, with the line running from the endurance limit to the ultimate strength.

Hooke's law

In the elastic region, stress is proportional to strain, $\sigma = E\varepsilon$.

Mach number

The ratio of flow speed to the local speed of sound; subsonic below 1, supersonic above 1.

Martensite

A hard, brittle non-equilibrium phase formed by rapidly quenching austenite.

Method of joints

A truss-analysis technique applying $\sum F_x = 0$ and $\sum F_y = 0$ at each pin joint.

Method of sections

A truss-analysis technique that cuts the truss and applies equilibrium to one part to find a specific interior member force directly.

Modulus of elasticity

Young's modulus E — the slope of the elastic region of the stress-strain curve; a measure of stiffness.

Mohr's circle

A graphical method giving the principal stresses and the maximum in-plane shear stress for a 2D stress state.

Moment of inertia

The second moment of area about an axis ($I = \int y^2\,dA$); a measure of a section's resistance to bending and buckling.

Natural frequency

The frequency at which a system oscillates freely, $\omega_n = \sqrt{k/m}$ for a spring-mass system.

Newton's second law

The net force on a particle equals its mass times acceleration, $\sum F = ma$; the foundation of kinetics.

Normal stress

Internal force per unit area perpendicular to a cross-section; for axial load $\sigma = P/A$.

Pearlite

A lamellar ferrite-and-cementite microstructure formed by slow cooling of eutectoid austenite.

Percent elongation

A ductility measure — the change in gauge length over the original gauge length, times 100.

PID controller

A feedback controller summing proportional (present error), integral (past error), and derivative (rate of change) actions.

Principal stresses

The extreme normal stresses, acting on planes where the shear stress is zero.

Professional Engineer (PE)

A licensed engineer authorized to offer services to the public and seal engineering work; FE passage plus experience and the PE exam earns the license.

Rankine cycle

The ideal steam power cycle: isentropic pump, constant-pressure boiler, isentropic turbine, constant-pressure condenser.

Resonance

The large-amplitude response when a forcing frequency approaches a system's natural frequency.

Reynolds number

A dimensionless ratio of inertial to viscous forces, $Re = \rho V D/\mu$; predicts laminar (< 2300) vs. turbulent (> 4000) pipe flow.

Second law of thermodynamics

Heat flows spontaneously hot-to-cold, no cyclic engine is 100% efficient, and entropy increases in real processes.

Shear stress

Internal force per unit area parallel to a cross-section.

Static equilibrium

The condition in which the net force and net moment on a body are zero ($\sum F = 0$, $\sum M = 0$); the basis of all statics analysis.

Three modes of heat transfer

Conduction (molecular contact), convection (fluid motion), and radiation (electromagnetic waves, needing no medium).

Transfer function

The ratio of the Laplace transform of a system's output to its input, with zero initial conditions.

Two-force member

A member loaded at only two points; in equilibrium its force must act along the line joining the two points.

Ultimate tensile strength

The maximum stress on the engineering stress-strain curve, before necking and fracture.

Von Mises (distortion-energy) theory

The most accurate static-failure theory for ductile metals; yielding occurs when the von Mises stress reaches the yield strength.

Work-energy theorem

The net work done on a particle equals its change in kinetic energy.

Yield strength

The stress marking the onset of permanent deformation; found by the 0.2% offset method when there is no sharp yield point.

FE Mechanical Study Guide FAQ

References

1. 1.NCEES. “FE Mechanical CBT Exam Specifications (effective July 2020).” ncees.org, 2020. ↑
2. 2.NCEES. “FE Exam — Format, Length, and Computer-Based Testing.” ncees.org. ↑
3. 3.NCEES. “FE Reference Handbook.” ncees.org. ↑
4. 4.NCEES. “Model Law and Model Rules — Rules of Professional Conduct.” ncees.org. ↑
5. 5.National Institute of Standards and Technology (NIST). “SI Units and the Metric System.” nist.gov. ↑

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