FE Mechanical exam practice: practice problems and exam questions

Practice problems for the Fundamentals of Engineering (FE) Mechanical exam: statics, mechanics of materials, dynamics and vibrations, thermodynamics and power cycles, fluid mechanics, heat transfer and machine design.

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Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

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Topics in this course

  1. Beam support reactions1 point load · 2 point loads · point load + UDLcalculations
  2. Axial stress & strainnormal stress · strain and elongation · stepped barcalculations
  3. Torsion of shaftsshear stress · angle of twist · power transmissioncalculations
  4. Shear force & bending momentpoint load · full-span UDL · point load + UDLcalculations
  5. Mohr’s circle & stress transformationprincipal stresses · maximum shear and principal direction · stress on an inclined planecalculations
  6. Beam deflectioncantilever, end load · uniform load · superposition and designcalculations
  7. Column bucklingpinned ends · other end conditions · design and critical stresscalculations
  8. Projectile motionhorizontal launch · launch from ground · launch from heightcalculations
  9. Work & energyfree fall · slope with friction · ramp, friction and springcalculations
  10. Mechanical vibrationsnatural frequency · damping ratio · springs in series and parallelcalculations
  11. Heat engines & refrigeratorsfirst law of a cycle · Carnot limit · refrigerators and heat pumpscalculations
  12. Gas power cyclesOtto cycle · Diesel cycle · Brayton cyclecalculations
  13. Continuity & Bernoullicontinuity · horizontal reducer · rising pipecalculations
  14. Pipe friction & head lossReynolds number · Darcy–Weisbach head loss · minor losses and total headcalculations
  15. Pumps & pump powerhydraulic power · pump efficiency · system head and powercalculations
  16. Heat conductionsingle wall · composite wall · wall with convectioncalculations
  17. Gears & gear trainsspeed ratio · torque and power · compound gear traincalculations
  18. Laplace transformstable pairs and value theorems · inverse transform by partial fractions · ODEs and transfer functionscalculations
  19. Block diagramsseries, parallel and feedback · closed-loop transfer functions and errors · disturbances, stability and polescalculations
  20. Step responsefirst-order systems · second-order parameters · physical systems and designcalculations
  21. Steady-state error & PID controlsystem type and error constants · steady-state errors and gain · P, PI and PD designcalculations

Exam tips for FE Mechanical exam practice

  • Check units before anything else: most wrong answers on the FE are unit slips.
  • Use the reference handbook’s standard cases for beam deflection instead of integrating.
  • In thermodynamics, always convert temperatures to kelvin.

Worked examples

Beam support reactionsEasy
P = 55 kNABRA​RB​ = ?2 m2 m

The simply supported beam has a pin at A and a roller at B. Find the vertical support reaction RB.

L = 4 mP = 55 kN at 2 m

Find RB (kN)

Show hints
  1. Use static equilibrium: ΣFy = 0 and ΣM = 0. Two unknowns, two equations.
  2. Take moments about A. RA passes through A, so it drops out and you get RB directly.
  3. Total downward load is 55 kN, so RA + RB = 55 kN.
Show the answer and the step-by-step solution

Static equilibrium: a body at rest has ΣFx = 0, ΣFy = 0 and ΣM = 0 about any point. Taking moments about one support removes that support’s reaction from the equation.

P = 55 kNABRA​RB​2 m2 m

Free-body diagram

Replace the supports with reaction forces. A roller gives one vertical reaction; with only vertical loads the pin gives one vertical reaction too.

Equilibrium

Moments about A, counter-clockwise positive. Then vertical force balance.

ΣMA = 0: RB·4 = 55·2
RB = 110/4 = 27.5 kN
ΣFy = 0: RA = 55 − 27.5 = 27.5 kN

Answer

RB = 27.5 kN
Axial stress & strainMedium
Ø 20 mmBrass2.5 mP = 10 kN

Find the axial strain ε in the brass bar, in microstrain (µε = 10⁻⁶).

P = 10 kNd = 20 mmL = 2.5 mE = 100 GPa

Find ε (µε)

Show hints
  1. Chain three relations: σ = P/A, ε = σ/E, δ = ε·L.
  2. Convert E to MPa (GPa × 1000) so it matches stress in MPa, and L to mm.
  3. σ = 31.83 MPa.
Show the answer and the step-by-step solution

Hooke’s law: within the elastic range σ = E·ε. Combining with σ = P/A and ε = δ/L gives δ = PL/(AE).

Stress

A = π·20²/4 = 314.2 mm²
σ = P/A = 10·1000/314.2 = 31.83 MPa

Strain (Hooke’s law)

E for brass = 100 GPa = 100000 MPa.

ε = σ/E = 31.83/100000 = 0.0003183 = 318.3 µε

Answer

ε = 318.3 µε

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Inverse Laplace transform calculator (partial fractions)Laplace transform calculatorBeam reaction calculatorBeam deflection calculatorShear force and bending moment diagram calculatorShaft torsion calculatorAxial stress and strain calculatorEuler buckling calculatorMohr’s circle calculatorGear ratio calculatorProjectile motion calculatorBernoulli equation calculatorHeat conduction calculatorCarnot efficiency and COP calculatorPipe head loss calculator (Darcy–Weisbach)Pump power calculatorNatural frequency and damping calculatorOtto and Brayton cycle efficiency calculatorBending stress calculator (σ = My/I)Reynolds number calculatorThermal expansion calculatorBelt length calculator (open and crossed belts)Kinetic energy calculator (and potential energy, work and power)

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