Reference
Engineer's Handbook
The formulas that run mechanical engineering — beams, sections, torsion, buckling, vessels, heat, vibrations, gears, and more. Classical results restated in our own words on a public-domain foundation.
Beam deflections & moments
Euler–Bernoulli results for the canonical support/load cases. δ = max deflection, M = max bending moment; E·I is the flexural rigidity. Valid when shear deflection is negligible (span/depth ≳ 10).
A load bends the beam into a curve; the ends stay put and the middle sags — that sag is the deflection δ.
Simply supported · point load P at midspan
δ = P·L³ / 48·E·I M = P·L / 4
Deflection at midspan; moment peaks under the load.
Simply supported · uniform load w
δ = 5·w·L⁴ / 384·E·I M = w·L² / 8
w in force per unit length; moment peaks at midspan.
Cantilever · point load P at tip
δ = P·L³ / 3·E·I M = P·L
16× the midspan-loaded simply supported deflection — cantilevers are soft.
Cantilever · uniform load w
δ = w·L⁴ / 8·E·I M = w·L² / 2
Moment peaks at the wall; that is where cantilevers crack.
Fixed–fixed · point load P at midspan
δ = P·L³ / 192·E·I M = P·L / 8
Clamping both ends cuts midspan deflection 4× vs simply supported.
Fixed–fixed · uniform load w
δ = w·L⁴ / 384·E·I M = w·L² / 12 (ends)
End moments govern; the wall moment is 1.5× the midspan moment.
Bending stress (any beam)
σ = M·c / I
c = distance from neutral axis to the extreme fiber (h/2 for symmetric sections).
Curvature from moment
1/R = M / (E·I)
The fundamental beam equation: bending moment M curves the beam, and E·I is the flexural rigidity. Integrate it twice with the end conditions and you recover slope and deflection.
Transverse shear stress
τ = V·Q / (I·b); τ_max = 1.5·V/A (rectangle), 4·V/3·A (solid circle)
Shear force produces a shear stress that peaks at the neutral axis, not the surface — why short, deep beams fail in shear rather than bending.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Section properties
Second moment of area I (bending stiffness), section modulus S = I/c (strength), area A. Depth wins: I grows with the cube of h.
The taller a section stands relative to bending, the stiffer it is — depth matters most.
Solid rectangle b × h
A = b·h I = b·h³ / 12 S = b·h² / 6
The workhorse; doubling depth gives 8× stiffness, 4× strength.
Solid circle, diameter d
A = π·d² / 4 I = π·d⁴ / 64 S = π·d³ / 32
Same in every bending direction — shafts and pins.
Hollow circle (tube), D outer, d inner
I = π·(D⁴ − d⁴) / 64
Material far from the axis does the work — tubes beat rods per kilogram.
Thin-walled tube, radius r, wall t
A ≈ 2π·r·t I ≈ π·r³·t
Valid for t ≪ r; the basis of aerospace semi-monocoque thinking.
Radius of gyration
r_g = √(I / A)
Feeds column buckling: slenderness = K·L / r_g.
Parallel-axis theorem
I = I_c + A·d²
Shift a section's second moment from its own centroid to a parallel axis a distance d away. It is how you build up I for I-beams, boxes and any composite section.
Polar second moment J
solid circle J = π·d⁴/32; J = I_x + I_y
The 'I' for torsion. For any section the polar moment equals the sum of the two bending second moments.
Section modulus Z
Z = I / c; σ = M / Z
Bundles I and the distance c to the outer fibre into one number, so bending stress is simply moment over section modulus. Bigger Z, lower stress.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Torsion of shafts
Circular shafts under torque T: shear stress peaks at the surface, twist accumulates along the length. G is the shear modulus.
A twisting moment winds the shaft; the far end rotates while the fixed end holds.
Shear stress, solid circular shaft
τ = T·r / J J = π·d⁴ / 32
J is the polar moment; τ is zero at the center, max at the surface.
Shear stress, tube
J = π·(D⁴ − d⁴) / 32
Same form; tubes carry torque efficiently.
Angle of twist
φ = T·L / (G·J)
Radians; the torsional analogue of δ = PL/EA.
Power–torque relation
P = T·ω = 2π·n·T / 60
n in rev/min, P in W with T in N·m — sizing every drive shaft since steam.
Torsional stiffness
T/θ = G·J / L
Torque per radian of twist — the torsional analogue of a spring rate. G is the shear modulus, J the polar moment, L the length.
Thin closed tube (Bredt)
τ = T / (2·A_m·t)
For a thin-walled closed section, shear stress depends on the area A_m enclosed by the wall mid-line — why a closed tube is vastly stiffer in torsion than the same tube slit open.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Column buckling (Euler)
Long columns fail by buckling long before the material yields. The critical load depends on stiffness and length squared — not strength.
Push a slender column past its critical load and it bows out between its pinned ends — buckling.
Euler critical load
P_cr = π²·E·I / (K·L)²
Above P_cr the straight form is unstable; slender parts bow sideways.
Effective length factors K
pinned–pinned 1.0 · fixed–free 2.0 · fixed–pinned 0.7 · fixed–fixed 0.5
End restraint is worth a lot: clamping both ends quadruples P_cr vs pinned.
Validity
slenderness K·L / r_g ≳ ~100 (steel-like)
Short columns crush instead — use material strength; intermediate columns need empirical curves.
Slenderness ratio
λ = L_e / r (L_e = K·L)
The single number that decides how a column fails: long and thin (high λ) buckles elastically; short and stubby (low λ) crushes. r is the radius of gyration.
Euler critical stress
σ_cr = π²·E / λ²
The Euler load written as a stress. It depends only on E and slenderness, not on strength — so a stiffer or stubbier column resists buckling, but a stronger material does not.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Thin-walled pressure vessels
Membrane stresses for internal pressure p, mean radius r, wall t. Valid for t/r ≲ 0.1.
Internal pressure pushes the wall outward; the ring stress resisting the split is the hoop stress.
Cylinder, hoop (circumferential)
σ_θ = p·r / t
Governs — twice the axial stress. Sausages split lengthwise for this reason.
Cylinder, axial
σ_z = p·r / 2·t
Half the hoop stress; end caps feel this.
Sphere
σ = p·r / 2·t
Equal-biaxial and minimal — why tanks for extreme pressure are spheres.
Required wall for safety factor SF
t = SF·p·r / σ_allow (cylinder)
Sphere needs half. Openings and nozzles need local reinforcement beyond this.
Hoop is twice axial
σ_hoop = p·r/t = 2·σ_axial
In a thin pressurised cylinder the circumferential (hoop) stress is double the lengthwise stress — why a cooking sausage splits along its length, not around it.
Thick-wall cylinder (Lamé)
σ_θ(bore) = p·(r_o² + r_i²) / (r_o² − r_i²)
When the wall is not thin (t ≳ D/10), stress varies through the wall and peaks at the bore. Use Lamé's equations, not the thin-wall p·r/t.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Heat transfer
The three ways heat moves, and the thermal Ohm's law that makes them calculable. k in W/(m·K), h in W/(m²·K), T in kelvin for radiation.
Heat flows from the hot end to the cold end until the temperature evens out.
Conduction (Fourier)
q = k·A·ΔT / L
Heat flows down the temperature hill through area A, path length L.
Thermal resistance
R = L / (k·A) ΔT = q·R
Ohm's law for heat: resistances in series add — walls, insulation, interfaces.
Convection (Newton cooling)
q = h·A·(Ts − T∞)
Typical h: still air 5–25, forced air 25–250, water 100–10 000 W/(m²·K).
Radiation (Stefan–Boltzmann)
q = ε·σ·A·(T₁⁴ − T₂⁴)
σ = 5.67×10⁻⁸ W/(m²·K⁴); the T⁴ makes it dominate at high temperature.
Lumped-capacitance time constant
τ = ρ·V·c / (h·A)
Valid when Biot = h·L/k < 0.1 — small or conductive bodies heat uniformly.
Why thick laminates cook
k_polymer ≈ 0.2 vs k_aluminum ≈ 237 W/(m·K)
Polymers are insulators: cure exotherm gets trapped — see the cure kinetics simulator.
Thermal expansion
ΔL = α·L·ΔT
Materials grow with temperature; α is the coefficient of thermal expansion (per °C). Small, but over a long span or a big ΔT it moves millimetres — the reason for expansion joints.
Thermal stress (constrained)
σ = E·α·ΔT
Stop a part expanding and the strain it wanted becomes stress instead. Mismatched expansion — carbon fibre against aluminium, say — cracks joints as they heat and cool.
Biot number
Bi = h·L_c / k
Decides whether the lumped-capacitance shortcut is valid: Bi < 0.1 means the part heats through almost uniformly; larger and you must treat the internal gradient.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Vibrations
Natural frequencies and damping — every structure rings at its own note, and resonance is how small forces do big damage.
The spring stretches and recoils as the mass oscillates at its natural frequency.
Spring–mass natural frequency
f = (1/2π)·√(k/m)
The master formula: stiffer → higher pitch, heavier → lower.
Cantilever beam, first mode
f₁ = (1.875² / 2π)·√(E·I / (ρ·A·L⁴))
ρ·A = mass per unit length; halving L quadruples the frequency.
Simply supported beam, first mode
f₁ = (π/2)·√(E·I / (ρ·A·L⁴))
Same physics, different boundary constant.
Damping via log decrement
δ = ln(xₙ / xₙ₊₁) = 2πζ / √(1−ζ²)
Measure two successive peaks, get the damping ratio ζ.
Resonance avoidance
keep forcing f < 0.7·fₙ or > 1.4·fₙ
Inside that band, dynamic amplification takes over — the classic design rule.
Polymers damp
damping ∝ tan δ
The DMA simulator's loss tangent is literally the material's vibration absorber.
Damped natural frequency
ω_d = ω_n·√(1 − ζ²)
Damping (ratio ζ) lowers the frequency slightly and bleeds energy each cycle. For light damping the shift is tiny, which is why the undamped ω_n is usually good enough.
Vibration isolation
isolate when f / f_n > √2 ≈ 1.41
A mount only isolates above √2 times its own natural frequency — below that it amplifies. Soft mounts (low f_n) isolate more, which is why engines sit on soft rubber blocks.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Fasteners & joints
Bolted and pinned joints: preload, shear, bearing — where most structures actually fail.
Tightening the bolt clamps the plates together with a steady preload force.
Torque–preload
T ≈ K·F·d
K ≈ 0.2 for dry steel threads; most of the torque fights friction, not stretch.
Bolt tension
σ = F / At
At = tensile stress area (from thread tables) — smaller than the nominal shank area.
Shear in a pin or bolt
τ = F / A (single) · F / 2A (double shear)
Double-shear joints halve the bolt stress — sandwich the lug.
Bearing stress
σb = F / (d·t)
The hole edge crushing under the bolt — governs thin sheets and laminates.
Composite joints knock down
bolted laminate joint ≈ 40–60% of laminate strength
Bearing/bypass interaction; why aerospace loves bonded joints and pad-ups.
Thread tensile stress area
A_t = (π/4)·(d − 0.938·p)²
A bolt breaks across its threads, whose effective area A_t is smaller than the shank — use A_t, not the nominal diameter, for tensile capacity. p is the thread pitch.
Preload target
F_i ≈ 0.75·A_t·σ_proof
A good bolted joint is tightened to about 75% of proof load, so it clamps hard and the bolt sees little of the fluctuating external load — the key to fatigue-proof joints.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Gears & drives
Ratios, torque multiplication, and the geometry that meshes. Speed down = torque up, minus friction's cut.
Meshing gears turn opposite ways; the smaller gear turns faster in proportion to its tooth count.
Gear ratio
i = ω₁/ω₂ = z₂/z₁ = d₂/d₁
Tooth counts set it exactly — gears never slip, belts do.
Torque multiplication
T₂ = i·T₁·η
η per mesh ≈ 0.98–0.99 spur/helical; worm drives 0.5–0.9.
Module (metric gearing)
m = d / z center distance a = m·(z₁+z₂)/2
Meshing gears must share the same module.
Pitch-line velocity
v = π·d·n / 60 000
d in mm, n in rev/min, v in m/s — drives noise, lubrication, and wear.
Belt drives
i ≈ d₂/d₁ (minus creep ~1–2%)
Friction-limited; V-belts wedge for grip, synchronous belts add teeth.
Pitch diameter
d = m·N
A metric gear's pitch diameter is its module m times its number of teeth N — the diameter of the imaginary circle on which two meshing gears effectively roll.
Centre distance
C = (d₁ + d₂)/2 = m·(N₁ + N₂)/2
How far apart to place two meshing gear shafts: the average of their pitch diameters. Get it wrong and the teeth bind or rattle.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Springs
Helical compression springs: rate from wire torsion, stress corrected for curvature. G is the shear modulus (steel ≈ 79 GPa).
Force rises in step with compression — the spring rate, k. The base stays fixed as the top is pressed down.
Spring rate
k = G·d⁴ / (8·D³·n)
d wire diameter, D mean coil diameter, n active coils — D³ makes coil diameter dominate.
Shear stress under load F
τ = Kw·8·F·D / (π·d³)
Wahl factor Kw ≈ (4C−1)/(4C−4) + 0.615/C, C = D/d — curvature concentrates stress inside the coil.
Springs in series / parallel
series: 1/k = Σ1/kᵢ · parallel: k = Σkᵢ
Opposite of resistors — soft springs dominate a series stack.
Surge frequency
f = (d / (2π·n·D²))·√(G/(2ρ))
Valve springs must stay well above the cam's forcing frequency.
Spring index
C = D / d
The ratio of coil diameter D to wire diameter d. Most springs sit between 6 and 12: too low and it is hard to wind and highly stressed; too high and it tangles and buckles.
Energy stored
U = ½·k·x²
A spring stores energy as the square of its deflection — the area under its force–deflection line. Double the deflection holds four times the energy.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Tolerances & fits
ISO limits and fits: the shorthand that decides whether parts slide, locate, or grip. Tolerance costs double roughly every grade tightened.
The shaft slides into the bore; the gap left around it is the clearance the fit allows.
Clearance fit
H7/g6 · H8/f7
Shaft always smaller than hole: bearings sliding, shafts rotating freely.
Transition fit
H7/k6 · H7/n6
Might touch, might clear: location fits for dowels and hubs assembled by hand or light press.
Interference fit
H7/p6 · H7/s6
Shaft always larger: press and shrink fits transmit torque by friction alone.
Surface roughness Ra, typical
machined 1.6–6.3 · ground 0.4–1.6 · lapped < 0.1 µm
Sealing faces and fatigue-critical surfaces want the low end.
The cost rule
each tighter grade ≈ 2× manufacturing cost
Tolerance only what the function needs — the cheapest design decision there is.
Tolerance stack-up
worst case Σδ ; statistical (RSS) √(Σδ²)
Adding tolerances along an assembly: worst-case sums them all (safe, pessimistic); RSS adds them in quadrature (realistic when the errors are independent and random).
Interference & pressure
p ∝ δ·E / d (shrink / press fit)
A press or shrink fit's grip comes from the interference δ squeezed between the parts, which sets up a contact pressure via Lamé. Too little and it slips; too much and it cracks the hub.
Classical results; tabulated since Kent's Mechanical Engineer's Pocket-Book (1895–1916, public domain).
Constants & quick values
The numbers that keep appearing.
A pendulum swings about its pivot, its period set by gravity g.
Standard gravity
g = 9.80665 m/s²
Exact by definition; 1 kgf = 9.80665 N.
Gas constant
R = 8.314 J/(mol·K)
Drives every Arrhenius rate — including our cure kinetics.
Standard atmosphere
1 atm = 101.325 kPa = 14.696 psi
Exact by definition.
Steel quick numbers
E ≈ 210 GPa · ρ ≈ 7.85 g/cm³
The benchmark composites are judged against (aluminum: 70 GPa, 2.70).
Approximate isotropy check
G = E / 2(1 + ν)
Holds only for isotropic materials — laminates that satisfy it are quasi-isotropic.
Quick unit conversions
1 MPa ≈ 145 psi; 1 GPa ≈ 145 ksi; °F = 1.8·°C + 32
The conversions you reach for constantly when moving between the metric units used here and US customary strength units.
Common densities (g/cm³)
steel 7.85 · aluminium 2.70 · titanium 4.5 · CFRP ~1.6 · water 1.00
Rough densities for quick weight estimates. Carbon composite comes in at roughly a fifth of steel — the whole point of using it.
Everyday moduli (GPa)
steel ~200 · aluminium ~70 · CFRP ~130 (along fibre) · glass/epoxy ~40 · nylon ~3
Ballpark stiffnesses: steel is three times aluminium, and a nylon part is nearly a hundred times floppier than a steel one of the same shape.
Defined values (SI) and CODATA.