Mechanics of Materials: practice problems and exam questions

Also called strength of materials: stress, strain and deformation of bars, shafts, beams and columns. Exam problems combine a statics step (the internal forces) with a materials step (stress or deflection), so a mistake in the first step carries through.

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

Topics in this course

  1. Axial stress & strainnormal stress · strain and elongation · stepped barcalculations
  2. Torsion of shaftsshear stress · angle of twist · power transmissioncalculations
  3. Shear force & bending momentpoint load · full-span UDL · point load + UDLcalculations
  4. Centroid & second moment of areacentroid · second moment of area · bending stresscalculations
  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

Exam tips for Mechanics of Materials

  • Keep units consistent: newtons and millimetres give stresses in MPa directly.
  • In torsion, use the radius, not the diameter, in τ = T·r/J.
  • For buckling, check the end conditions: the effective length can change the critical load by a factor of 16.

Worked examples

Axial stress & strainEasy
300 kNØ30120 GPaØ100/50500 mmσr​ = ?

A steel rod (diameter 30 mm, E = 200 GPa) sits inside a tube made of copper (outer diameter 100 mm, inner diameter 50 mm, E = 120 GPa). Both are 500 mm long and are compressed together between rigid plates by 300 kN. Find the compressive stress σr in the rod.

P = 300 kNrod: Ø30 mm, E = 200 GPatube: Ø100/Ø50 mm, E = 120 GPaL = 500 mm

Find σr (MPa)

Show hints
  1. Rigid plates: rod and tube shorten by the same amount, so ε is the same in both.
  2. ε = P/ΣEA.
  3. σ = E·ε.
Show the answer and the step-by-step solution

Two materials: in series (one after the other) every segment carries the same force and the elongations add: δ = P·Σ L/(E·A). In parallel (side by side between rigid plates) both parts shorten by the same amount, so their strains are equal: the stiffer part takes the larger share, Pi = P·EiAi/ΣEA, and σi = Ei·ε. A bar held between two rigid walls with a load at the junction is statically indeterminate: the two segments share the load in proportion to their stiffness k = EA/L.

Areas

Ar = π·30²/4 = 706.9 mm²
At = π(100² − 50²)/4 = 5890 mm²

Combined axial stiffness

ΣEA = 200·10³·706.9 + 120·10³·5890 = 8.482 × 10⁸ N

Common strain

ε = 300000/8.482 × 10⁸ = 0.0003537

Stress in the rod

σr = 200·10³·0.0003537 = 70.74 MPa

Answer

σr = 70.74 MPa
Torsion of shaftsMedium
T = 750 N·md = 40 mmL = 1.5 mφ = ?

A solid brass shaft is fixed at the wall and twisted by a torque T at its free end. Find the angle of twist φ of the free end, in degrees.

T = 750 N·md = 40 mmL = 1.5 mG = 39 GPa

Find φ (°)

Show hints
  1. Use φ = TL/(GJ), which gives radians.
  2. Use N and mm throughout: T in N·mm, L in mm, G in MPa (GPa × 1000), J in mm⁴.
  3. J = 251327 mm⁴.
Show the answer and the step-by-step solution

Torsion of circular shafts: shear stress grows linearly from zero at the centre to τmax = T·r/J at the surface, with J = πd⁴/32 for a solid shaft and J = π(do⁴ − di⁴)/32 for a hollow one. The angle of twist is φ = TL/(GJ) in radians. Power and torque are linked by P = T·ω.

Polar second moment of area

J = π·40⁴/32 = 251327 mm⁴

Angle of twist

φ = TL/(GJ) = 750·1000·1500/(39000·251327) = 0.1148 rad
φ = 0.1148·180/π = 6.576°

Answer

φ = 6.576°

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

Beam deflection calculatorShear force and bending moment diagram calculatorShaft torsion calculatorAxial stress and strain calculatorEuler buckling calculatorMohr’s circle calculatorMoment of inertia calculator (second moment of area)Bending stress calculator (σ = My/I)Thermal expansion calculatorCentroid calculator (composite shapes and polygons)

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