Validation
The calculation engines are tested against the 43 worked examples below. Each row gives the expected value, where it comes from and the value the engine computes. Each row is also an automated test: if the engine drifts from the expected value, the test fails and the change is not published.
The worked examples are part of 903 automated tests (counted 2026-09-30). Computed values are rounded to the precision each test asserts. Sources are as cited in the test files and on each calculator page. See the editorial policy for how sources are chosen and errors corrected.
Tolerance stack-up
Default stack: 25 ± 0.1 mm housing, less a 10 ± 0.05 mm spacer and a 5 ± 0.05 mm bush, all normal with k = 3. Open the tolerance stack-up calculator. Tests: src/calculators/tolerance-stack/engine.test.ts.
| Case and inputs | Expected | Source | Computed | Match |
|---|---|---|---|---|
| Equal-bilateral conversion 10 +0.2 / −0.1 mm | mean 10.050, ± 0.150 mm | Fischer, Mechanical Tolerance Stackup and Analysis, 2nd ed., 2011, equal-bilateral conversion | mean 10.050, ± 0.150 mm | Yes |
| Position with MMC bonus Position zone 0.2 mm, bonus 0.1 mm | ± 0.150 mm | ASME Y14.5-2018, bonus tolerance | ± 0.150 mm | Yes |
| Profile Profile zone 0.4 mm | ± 0.200 mm | ASME Y14.5-2018, profile zone | ± 0.200 mm | Yes |
| Uniform distribution ± 0.3 mm, uniform | σ = a / √3 = 0.17321 mm | Variance of a uniform distribution | 0.17321 mm | Yes |
| Worst case Default stack | 9.800 to 10.200 mm | Fischer, Mechanical Tolerance Stackup and Analysis, 2nd ed., 2011 | 9.800 to 10.200 mm | Yes |
| RSS Default stack | ± 0.122474 mm, σ 0.040825 mm | Fischer, Mechanical Tolerance Stackup and Analysis, 2nd ed., 2011 | ± 0.122474 mm, σ 0.040825 mm | Yes |
| RSS with Bender factor Default stack, factor 1.5 | ± 0.183712 mm, σ unchanged | Bender, A. (1962) "Benderizing tolerances: a simple practical probability method of handling tolerances for limit stack-ups", Graphic Science, December 1962 | ± 0.183712 mm, σ 0.040825 mm | Yes |
| Monte Carlo Default stack, seed 1, 20,000 samples | Mean 10 within 0.002 mm; σ within 3 % of RSS σ | Checked against the RSS result above | Mean 10.00005 mm; σ 0.041148 mm (0.8 % above RSS) | Yes |
| Monte Carlo yield Default stack, limits 9.9 to 10.1 mm | About 1.43 % outside (normal tails beyond ± 2.449 σ); test band 0.8 to 2.2 % | Normal distribution tail area | 1.51 % outside | Yes |
Cylinder force and sizing
Default cylinder: 63 mm bore, 36 mm rod (ISO 3320:2013 series), 200 bar, efficiency 0.9, 40 L/min, 500 mm stroke. Open the cylinder force and sizing calculator. Tests: src/calculators/cylinder/engine.test.ts.
| Case and inputs | Expected | Source | Computed | Match |
|---|---|---|---|---|
| Areas D 63 mm, d 36 mm | Piston 3117.245, rod 1017.876, annulus 2099.369 mm²; ratio 1.4848 | Hand calculation, A = π D² / 4 | 3117.245, 1017.876, 2099.369 mm²; 1.4848 | Yes |
| Forces Default cylinder | Extend 56,110.4 N, retract 37,788.6 N, regeneration 18,321.8 N | Hand calculation, F = p A η | 56,110.4 N, 37,788.6 N, 18,321.8 N | Yes |
| Back pressure Default cylinder, 10 bar on the annulus | Extend 54,221.0 N | Hand calculation, F = (p A − p_b A_a) η | 54,221.0 N | Yes |
| Pneumatic D 50 mm, d 20 mm, 6 bar, η 0.9 | Extend 1060.3 N, retract 890.6 N | Hand calculation, F = p A η | 1060.3 N, 890.6 N | Yes |
| Imperial D 2 in, d 1 in, 1000 psi, η 0.9 | Extend 2827.43 lbf, retract 2120.58 lbf | Hand calculation, F = p A η | 2827.43 lbf, 2120.58 lbf | Yes |
| Speeds and times Default cylinder | Extend 213.86, retract 317.56, regeneration 654.96 mm/s; 2.338 s out, 1.575 s back | Hand calculation, v = Q / A | 213.86, 317.56, 654.96 mm/s; 2.338 s, 1.575 s | Yes |
| Swept volumes Default cylinder | Cap side 1.5586 L, rod side 1.0497 L | Hand calculation, V = A L | 1.5586 L, 1.0497 L | Yes |
| Fluid power 200 bar, 40 L/min | 13,333.3 W | Hand calculation, P = p Q | 13,333.3 W | Yes |
| Bore sizing, extend 50 kN at 160 bar, η 0.9 | Minimum bore 66.49 mm, next ISO 3320 bore 80 mm; 178.22 bar needed at 63 mm | Hand calculation; ISO 3320:2013 bore series | 66.49 mm, 80 mm, 178.22 bar | Yes |
| Bore sizing, retract 50 kN at 160 bar, rod 36 mm | Minimum bore 75.61 mm, next bore 80 mm; 264.63 bar needed at 63 mm | Hand calculation; ISO 3320:2013 bore series | 75.61 mm, 80 mm, 264.63 bar | Yes |
| Euler buckling Rod 36 mm, L 1000 mm, K 1.0, E 210 GPa, load 56,110.4 N | I 82,448 mm⁴; F_cr 170,883 N; safety factor 3.045, below 3.5 so flagged | Euler column formula, Shigley's Mechanical Engineering Design, 10th ed. | 82,448 mm⁴; 170,883 N; 3.045, flagged | Yes |
| Mounting factor As above, K 0.5 | F_cr four times larger, 683,532 N; passes | Euler column formula, Shigley's Mechanical Engineering Design, 10th ed. | 683,532 N; passes | Yes |
Press and interference fit
Default fit: 50 mm H7/p6, solid steel shaft, steel hub OD 100 mm, length 40 mm, μ 0.15, E 210 GPa, ν 0.3, clearance 0.02 mm. Open the press and interference fit calculator. Tests: src/calculators/press-fit/engine.test.ts and iso286.test.ts.
| Case and inputs | Expected | Source | Computed | Match |
|---|---|---|---|---|
| Limit deviations 50 mm H7 and p6 | Hole +25 / 0 µm, shaft +42 / +26 µm | ISO 286-2:2010 tables | +25 / 0 µm, +42 / +26 µm | Yes |
| Tolerance grades IT5 to IT11 at 50 mm | 11, 16, 25, 39, 62, 100, 160 µm | ISO 286-1:2010 Table 1 | 11, 16, 25, 39, 62, 100, 160 µm | Yes |
| Interference Default fit | Min 1, mean 21.5, max 42 µm | From the ISO 286 limits | 1, 21.5, 42 µm | Yes |
| Contact pressure Default fit | 1.575, 33.8625, 66.150 MPa | Shigley's Mechanical Engineering Design, 10th ed., eq. 3-57 | 1.575, 33.8625, 66.150 MPa | Yes |
| Dissimilar materials Steel shaft in aluminium hub (70 GPa, ν 0.33), δ 42 µm | 26.368 MPa | Shigley's Mechanical Engineering Design, 10th ed., general Lamé form | 26.368 MPa | Yes |
| Hollow shaft Shaft bore 30 mm, δ 42 µm | 46.523 MPa | Shigley's Mechanical Engineering Design, 10th ed., eq. 3-56 with equal materials | 46.523 MPa | Yes |
| Stresses at max interference Default fit | Hub hoop 110.250, hub von Mises 154.350, shaft −66.150 MPa | Lamé thick cylinder, Shigley's Mechanical Engineering Design, 10th ed. | 110.250, 154.350, −66.150 MPa | Yes |
| Hollow shaft stresses p 40 MPa, d 50 mm, bore 30 mm | Hoop −85 MPa at surface, −125 MPa at bore | Lamé thick cylinder, Shigley's Mechanical Engineering Design, 10th ed. | −85, −125 MPa | Yes |
| Holding force and torque Default fit | 1.484, 31.915, 62.345 kN; 37.11, 797.87, 1558.62 N·m | Hand calculation, F = μ p π d L, T = F d / 2 | 1.484, 31.915, 62.345 kN; 37.11, 797.87, 1558.62 N·m | Yes |
| Assembly temperature Default fit, α 11.5 × 10⁻⁶ /K, from 20 °C | ΔT 107.826 K, hub at 127.826 °C | Hand calculation, ΔT = (δ + c) / (α d) | 107.826 K, 127.826 °C | Yes |
| Default clearance Clearance blank, c = 0.001 d | ΔT 160 K | Hand calculation | 160 K | Yes |
| Smoothing loss Rz 4 and 6.3 µm | 8.24 µm | DIN 7190-1:2017, 0.8 (Rz_shaft + Rz_hub) | 8.24 µm | Yes |
| Thermal loss of fit Steel shaft in 6061 hub at 120 °C | −60.5 µm, fit lost | Hand calculation, Δδ = d (α_s − α_h) ΔT | −60.5 µm, fit lost | Yes |
In all, 33 limit deviations from the ISO 286-2:2010 tables, from 2 mm to 500 mm and covering hole letters F to S and shaft letters f to u, are checked in iso286.test.ts.
Bolt torque and preload
Default joint: M12 × 1.75, ISO 898-1 class 8.8 (proof 580 MPa, yield 640 MPa), 75 % of proof, K 0.20, μ 0.12 thread and head, d_w 16.6 mm, hole 13.0 mm. Open the bolt torque and preload calculator. Tests: src/calculators/bolt-torque/engine.test.ts.
| Case and inputs | Expected | Source | Computed | Match |
|---|---|---|---|---|
| Thread geometry M12 × 1.75 | d2 10.863, d3 9.853 mm; As 84.27 mm² (ISO 898-1 tabulates 84.3) | ISO 724 basic dimensions; ISO 898-1:2013 | 10.863, 9.853 mm; 84.27 mm² | Yes |
| Inch stress area 1/2-13 UNC | 0.1419 in² | ASME B1.1-2019, As = 0.7854 (d − 0.9743/n)² | 0.1419 in² | Yes |
| Proof load Default joint | 48.87 kN (ISO table 48.9 kN) | ISO 898-1:2013 | 48.87 kN | Yes |
| Target preload 75 % of proof | 36.66 kN | Hand calculation | 36.66 kN | Yes |
| Torque, method A K 0.20 | 87.97 N·m | Hand calculation, T = K F d | 87.97 N·m | Yes |
| Torque, method B μ 0.12 thread and head | 70.53 N·m; equivalent K 0.160; split 10.26 / 27.72 / 32.55 N·m | VDI 2230 Part 1:2015 | 70.53 N·m; 0.160; 10.26 / 27.72 / 32.55 N·m | Yes |
| Preload scatter 70.53 N·m, friction ± 20 % | 31.31 to 44.21 kN | VDI 2230 Part 1:2015 relation, inverted | 31.31 to 44.21 kN | Yes |
| Thread stress when tightening Default joint | σ 435.0, τ 174.0, von Mises 529.2 MPa; utilisation 0.827 | VDI 2230 Part 1:2015 | 435.0, 174.0, 529.2 MPa; 0.827 | Yes |
| SAE Grade 5 1/2-13 UNC, proof 85 ksi, 75 %, K 0.20 | Proof 12,061 lbf, preload 9,046 lbf, torque 904.6 lbf·in (75.38 lbf·ft) | SAE J429; ASME B1.1-2019 | 12,061 lbf, 9,046 lbf, 904.6 lbf·in (75.38 lbf·ft) | Yes |