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Mechanics of Materials

From transcript: 20ME340 Mechanics of Materials (E) · 20ME48L Basic Material Testing (A)

Cheat sheet

Printable study sheet: formulas → definitions → topic notes → traps.

Formulas

Axial: σ = P/A ; δ = PL/(AE) ; thermal δ = α L ΔT ; total δ = mechanical + thermal

Torsion (circular): τ/r = T/J = Gθ/L ; J_solid = π d⁴/64 ; J_hollow = π(Do⁴−Di⁴)/64

Bending: σ = M y / I ; max at outer fiber; neutral axis through centroid

Shear in beams: τ = V Q /(I b) ; rectangular max = 1.5 V/A at NA

Combined: Mohr’s circle; σ_{1,2} = σx/2 ± √((σx/2)²+τ²) for plane stress σy=0

Hooke: σ = E ε ; γ = τ/G ; ε_lat = −ν ε_long ; E = 2G(1+ν)

Strain energy axial U = P²L/(2AE) ; torsion U = T²L/(2GJ)

Euler buckling: Pcr = π² E I / (K L)² ; K pinned-pinned=1; fixed-free=2; fixed-fixed=0.5; fixed-pinned≈0.7

Slenderness λ = K L / k ; k=√(I/A)

Max shear theory (Tresca): τ_max ≤ Sy/2 ; von Mises: √(σ1²+σ2²+σ3²−σ1σ2−…) ≤ Sy

Fatigue Goodman: σa/Se + σm/Sut ≤ 1/n ; Soderberg uses Sy for mean

Stress concentration: σ_max = Kt σ_nom

Pressure vessel thin: hoop σθ = p r / t ; long. σz = p r /(2t)

σ = P/A ; ε = δ/L ; E = σ/ε (elastic)

δ = PL/(AE)

Bending: σ = My/I ; τ = VQ/(Ib)

Torsion (round): τ = Tr/J ; θ = TL/(GJ)

Combined: Mohr's circle — principal σ1,2 = σx/2 ± √((σx/2)²+τ²)

Euler buckling: Pcr = π²EI/(KL)²

Definitions

Elastic limit / yield: Onset of permanent set; Sy used in ductile design
Ultimate strength Sut: Max engineering stress on curve
Modulus E: Slope of elastic σ–ε ; stiffness of material
Poisson ν: −ε_lat/ε_long ; metals ~0.3
Section modulus Z: I/y_max — bending strength geometric measure
Principal stress: Normal stresses on planes of zero shear
Stiffness vs strength: Deflection resistance vs load to fail
Buckling: Sudden lateral instability of compressed slender members
Endurance limit Se: Stress amplitude for infinite life (steels often)
Factor of safety n: Strength / allowable working stress
Stress / strain: Internal force intensity / relative deformation
Yield strength: Onset of permanent deformation (design often uses Sy/n)

Topic-wise short notes

Study these first — one block per syllabus topic. Then read the deep notes below.

Axial, bending, torsion

Statically indeterminate axial: compatibility of δ + equilibrium.
Beams: draw V & M first; σ_b from M; deflection via EI y''=M.
Double integration, Macaulay, moment-area, energy (Castigliano) methods.
Unsymmetric bending: resolve about principal axes of inertia.
Shear center: point where load causes bending without twist.

Failure theories

Ductile → Tresca (conservative) or von Mises (better match).
Brittle → max normal stress theory.
Mohr’s theory for different tension/compression strengths.
Always state which theory and why material class.

Axial, torsion, thermal

Bars in series: same P, δ sum; parallel: same δ, P sum.
Composite shafts: θ same if joined; share T by GJ stiffness.

Beams & buckling

Boundary K factors dominate Pcr — fixity is design.
Intermediate columns: Rankine/Gordon empirical vs pure Euler.
Kt at fillets, keyways, holes — use charts; fatigue sensitive.

Failure theories & fatigue

Infinite life if σa below Se (adjusted); else finite life via S–N.
Mean stress tensile hurts fatigue; compressive mean often less harmful.
Miner's rule for variable amplitude cumulative damage (approx).

Pressure vessels & combined loading

Thin wall: hoop = pr/t, long = pr/(2t); thick wall Lame equations.
Combined axial+bending+torsion → Mohr or principals then failure theory.
Thermal stress when δ_thermal prevented: σ=EαΔT (fixed ends).

Exam traps & quick notes

Sign conventions matter for Mohr's circle.
Lab: tensile test → E, Sy, Sut, % elongation from stress–strain curve.
Project: BAJA chassis FEA validates beam/tube stress intuition.
Sign: tension +, compression − (course convention).
Superposition only in linear elastic range.
Thermal stress if expansion constrained: σ = E α ΔT (fully fixed).
Non-circular torsion: warping — don’t use simple J formulas blindly.
Short columns crush; long columns buckle — check both.
Fatigue: surface finish, size, reliability, stress concentration cut Se.

Comprehensive notes

Axial, bending, torsion

Superposition for combined loading; neutral axis; shear flow in thin walls.

External: MIT Mechanics of Materials ↗ Strength of materials (NPTEL) ↗

Failure theories

Max normal (brittle), Tresca & von Mises (ductile). Fatigue: S–N, endurance limit, Goodman/Soderberg.

External: efunda failure theories ↗

Axial, torsion, thermal

δ=PL/AE; thermal δ=αLΔT; torsion τ=Tr/J, θ=TL/GJ for circular shafts. Combined loading → principal stresses via Mohr.

External: NPTEL ↗ Engineering Toolbox ↗

Beams & buckling

σ=My/I; shear flow; deflection methods (double integration, energy). Euler buckling Pcr=π²EI/(KL)² with end-condition K. Stress concentrations Kt at fillets/holes.

External: NPTEL ↗ Engineering Toolbox ↗

Failure theories & fatigue

Ductile: Tresca/von Mises. Brittle: max normal. Fatigue: S–N, endurance limit, mean stress (Goodman). Design for infinite life vs finite life.

External: NPTEL ↗ Engineering Toolbox ↗

Interview Q&A for this subject

Q: Where is bending stress max in a cantilever with end load?
A: At the fixed root outer fibers — M is max at support.

External: ASQ / quality ↗ MIT OCW ↗