Limits, Fits and Tolerances: Method

The ISO System of Limits and Fits: Formulas, Rules and Conventions

Terms

A fit is the relation between a hole and a shaft of the same basic (nominal) size. Each part is given two limits of size; the differences between the limits and the basic size are the deviations, written ES and EI (upper and lower) for the hole and es and ei for the shaft; the tolerance is the difference between the two limits, T = ES − EI or es − ei. A tolerance class such as H7 or g6 names a fundamental deviation, the deviation nearest the zero line, by a letter (capital for holes, lower case for shafts) and a standard tolerance grade IT by a number. The hole-basis system keeps the hole at H (EI = 0) and obtains the fits by the shaft letter; the shaft-basis system keeps the shaft at h (es = 0). The allowance is the intentional difference between mating parts (TC 9-524, NAVEDTRA 12204): a clearance allowance when the minimum clearance is positive, an interference allowance when the shaft is larger than the hole, and the tool uses the sign convention clearance positive, interference negative.

Basic size steps

The standard tolerances and the fundamental deviations are not computed for each size but for steps of the basic size (ISO 286-1 Table 4): above 3 to 6, 6 to 10, 10 to 18, 18 to 30, 30 to 50, 50 to 80, 80 to 120, 120 to 180, 180 to 250, 250 to 315, 315 to 400 and 400 to 500 mm, with the first step up to and including 3 mm, and intermediate steps (10 to 14, 14 to 18, 18 to 24 … 450 to 500) used only for the deviations a, b, c and r to zc, whose values grow faster with size. Every formula is evaluated at the geometric mean D = √(D₁ D₂) of the step, taken between 1 and 3 mm for the first step.

Standard tolerances

The standard tolerance factor i = 0.45 ∛D + 0.001 D (micrometres, D in mm) was derived empirically on the premise that, for one manufacturing process, the error grows about as a parabola of the size. The grades are multiples of it: IT5 = 7i, IT6 = 10i, IT7 = 16i, IT8 = 25i, IT9 = 40i, IT10 = 64i, IT11 = 100i, IT12 = 160i, IT13 = 250i, IT14 = 400i, IT15 = 640i, IT16 = 1000i, IT17 = 1600i, IT18 = 2500i, a geometric progression of ratio about 1.6 that multiplies the tolerance by ten every fifth grade. The three finest grades have their own linear formulas, IT01 = 0.3 + 0.008 D, IT0 = 0.5 + 0.012 D, IT1 = 0.8 + 0.02 D, and IT2, IT3 and IT4 have no formula: the standard scaled them approximately in geometric progression between IT1 and IT5, which is what the tool does. Calculated values up to IT11 are rounded to 1 µm up to 100 µm, 5 µm up to 200 µm and 10 µm above; the grades above IT11 are then ten times the rounded grade five steps below, so they need no rounding of their own.

The standard says of its own table that "for the small values in particular, it has sometimes been necessary to depart from these rules" and that the tabulated values are definitive (ISO 286-1 Annex A, Table 8 note 1). The tool prints the formula values and does not reproduce the table; the page marks the IT01 to IT4 values as approximate, and an override field takes the table value when the analyst holds the standard and finds a cell that differs.

Fundamental deviations of shafts

ISO 286-1 Annex A, Table 9, gives the shaft deviations as functions of D in micrometres. The clearance letters have a negative upper deviation es: a = −(265 + 1.3 D) up to 120 mm and −3.5 D above, b ≈ −(140 + 0.85 D) up to 160 mm and −1.8 D above, c = −52 D^0.2 up to 40 mm and −(95 + 0.8 D) above, d = −16 D^0.44, e = −11 D^0.41, f = −5.5 D^0.41, g = −2.5 D^0.34, h = 0, with cd, ef and fg the geometric means of their neighbours (sizes up to 10 mm). The transition and interference letters have a positive lower deviation ei: k = +0.6 ∛D for grades IT4 to IT7 and 0 for the others, m = IT7 − IT6, n = +5 D^0.34, p = IT7 + 0 to 5, r = √(p s), s = IT8 + 1 to 4 up to 50 mm and IT7 + 0.4 D above, t = IT7 + 0.63 D (above 24 mm), u = IT7 + D, v = IT7 + 1.25 D (above 14 mm), x = IT7 + 1.6 D, y = IT7 + 2 D (above 18 mm), z = IT7 + 2.5 D, za = IT8 + 3.15 D, zb = IT9 + 4 D, zc = IT10 + 5 D. The symmetrical js has es = −ei = IT/2, and j has no formula at all (its values exist only in the tables, for grades 5 to 7). The results are rounded by the standard's Table 10: for a to g to 1 µm up to 45 µm, 2 µm to 60, 5 µm to 200, 10 µm to 560, 20 µm to 1000 and 50 µm above; for k to zc to 1 µm up to 100 µm, 2 µm to 300, 5 µm to 600, 10 µm to 800, 20 µm to 1000, 50 µm to 2000 and 100 µm above.

Two formulas are ranges, p = IT7 + 0 to 5 and s = IT8 + 1 to 4; the increment the standard chose for each size step is taken from its table as a fact (0 for p at 3 to 18 mm, 1 at 18 to 50, 2 at 50 to 120, 3 at 120 to 180, 4 at 180 to 315, 5 at 315 to 500; 1, 1, 1, 2, 4 for s from 3 to 50 mm; and p = +6 below 3 mm, which is below IT7), and j has no formula at all, so j5 to j7 are tabulated values; the page marks both "tabulated". Every other deviation is the formula value with the Table 10 rounding. The standard notes that its printed tables depart from the formulae in places (most of them in the loose letters a, b, c and below 18 mm in the heavy interference letters, where values were kept from the earlier ISA system); such a cell is entered in the override field.

Fundamental deviations of holes

The hole with a given letter is the mirror image of the shaft with the same letter: EI = −es for A to H and ES = −ei for K to ZC (ISO 286-1 A.4.2), so that H7/f6 and F7/h6 are the same fit. The rule is modified where a hole of grade n mates with a shaft of grade n − 1: to keep the hole-basis and shaft-basis fits identical in their clearances (H7/p6 and P7/h6, say), the hole deviations K, M and N up to IT8 and P to ZC up to IT7 take ES = −ei + Δ with Δ = ITn − IT(n − 1), the difference of the two tolerance grades; the standard's own example is P7 in the 18 to 30 mm step, Δ = IT7 − IT6 = 21 − 13 = 8 µm, giving ES = −22 + 8 = −14 µm. Three further rules of the standard: N above IT8 has ES = 0 (N9, N10, N11 are "hole-basis-like" with the whole tolerance below the zero line), K above IT8 is not defined above 3 mm, and M6 in the 250 to 315 mm step has ES = −9 instead of −11. For sizes up to 3 mm there is no Δ and the hole values are K = 0, M = −2, N = −4 and P = −6 for every grade. J, like j, has no formula (J6, J7, J8 are tabulated).

Limits and fit

The second deviation of each part follows from its tolerance, ES = EI + IT or EI = ES − IT for the hole and ei = es − IT or es = ei + IT for the shaft, and the limits of size are the basic size plus the deviations. The maximum clearance pairs the largest hole with the smallest shaft, C_max = ES − ei, and the minimum clearance the smallest hole with the largest shaft, C_min = EI − es; the fit is a clearance fit when C_min ≥ 0, an interference fit when C_max ≤ 0 (every shaft larger than every hole) and a transition fit between. The variation of the clearance, C_max − C_min = T_h + T_s, is the fit tolerance. Any designation is accepted by typing the hole and shaft classes; ANSI B4.2 lists a selection of preferred fits for general use (cited, not reproduced here). The page also tabulates the two letters in grades 5 to 11 at the entered size, which shows how the fit changes with the grade, and draws the tolerance-zone diagram about the zero line.

Inch fits

The ANSI B4.1 classes RC (running and sliding), LC (locational clearance), LT (locational transition), LN (locational interference) and FN (force and shrink) tabulate the hole and shaft limits in thousandths of an inch for size ranges of their own, with grades of their own; the table is not reproduced here. In the inch mode the four limits read from it are entered with the class named, and the tool computes the deviations from the basic size, the clearance extremes, the type, the diagram, the statistics and the position-tolerance section exactly as for an ISO fit.

Statistical clearance

The worst-case extremes pair the largest hole with the smallest shaft and the reverse. If instead each part comes from a centred process whose tolerance is k standard deviations (k = 3 for the usual ±3σ convention), the clearance is close to normal with mean (C_max + C_min)/2 and standard deviation σ_C = √(σ_h² + σ_s²), σ_h = T_h/(2k), σ_s = T_s/(2k): the two tolerances add as a root sum of squares, not arithmetically, and the ±3σ band is narrower than the worst case by the factor √(T_h² + T_s²)/(T_h + T_s). The probability of interference is Φ(−C_mean/σ_C) with Φ the standard normal distribution; for a transition fit it is the fraction of assemblies that will need a press. The Tolerance Stack-Up tool treats longer chains with uniform and triangular parts, mean shifts, Bender's factor and a Monte Carlo (Scholz 1995).

Position tolerance at MMC

When a hole or a pin carries a position tolerance stated at the maximum material condition (ASME Y14.5), the stated tolerance applies at MMC (the smallest hole, the largest shaft) and grows as the feature departs from it: bonus = |actual size − MMC|, up to the whole size tolerance at LMC. The virtual condition is the worst-case mating boundary that a functional gauge is made to, MMC − t for a hole and MMC + t for a shaft, and the resultant condition the extreme boundary on the LMC side, LMC + t + bonus for a hole and LMC − t − bonus for a shaft. With a position tolerance on both parts, the difference of the virtual conditions, (D_min − t_h) − (d_max + t_s), must not be negative for a hole and a shaft at MMC that each use their whole position tolerance to assemble; this is the fixed-fastener condition, and for a floating fastener (a clearance hole in each of two parts around one bolt) the same arithmetic is applied part by part.

Envelope requirement and surface finish

The Taylor principle, the envelope requirement of ISO 286-1 (clause 5.3.1.2) and Rule #1 of ASME Y14.5, states that a feature of size at its maximum material limit must be of perfect form: the mating envelope (the largest shaft, the smallest hole) is checked by a full-form GO gauge made to the MMC size, or to the virtual condition when a position tolerance applies, and the opposite limit by a NOT GO gauge at every local two-point size. The limits of size are measured on the real surface, whose peaks and valleys span several Ra, so a roughness that is not small against the tolerance makes the size itself ill defined and the fit variable; the finishing process is chosen with the grade (the GSFC drawing manual lists finish classes for rough machining, mating surfaces, moving parts, precision, polished and lapped parts; the symbols and definitions are ASME B46.1 and Y14.36). The page reports an entered Ra as a fraction of the tolerance; the roughness is not part of the ISO 286 limits.

Worked example on the page

The default is a 50 mm H7/g6 sliding fit. The 30 to 50 mm step has D = √(30 × 50) = 38.73 mm and i = 0.45 ∛38.73 + 0.0387 = 1.561 µm, so IT7 = 16 i = 24.98 → 25 µm and IT6 = 10 i = 15.61 → 16 µm. The H hole has EI = 0 and ES = +25 µm; the g shaft has es = −2.5 × 38.73^0.34 = −8.67 → −9 µm and ei = −9 − 16 = −25 µm. The limits are 50.025 / 50.000 mm and 49.991 / 49.975 mm, the clearance runs from 9 to 50 µm, mean 29.5 µm, a clearance fit; the RSS model with k = 3 gives σ_C = √((25/6)² + (16/6)²) = 4.95 µm and a 3σ band of 14.7 to 44.3 µm. The same arithmetic gives 25 mm H7/p6 as +21/0 against +35/+22 (p = IT7 + 1 in the 18 to 30 mm step, IT6 = 13), an interference of 1 to 35 µm, and the hole-basis rule reproduces P7/h6 at 25 mm as −14/−35 against 0/−13, the same interference. The test script checks the grades IT5 to IT11 at every size step, the deviations of g, h, k, p, s, u and the holes K, M, N, P at ten sizes, the H7/g6, H7/k6, H7/p6 and H7/s6 fits at 25 and 50 mm and the rounding rules against the printed tables of the standard (reporting, not hiding, the cells where the standard's own tables depart from its formulae), and a numpy re-implementation of the whole system.

Limits of the tool

• Basic sizes above 0 and up to 500 mm; the 500 to 3150 mm range of ISO 286 (a different tolerance factor and deviation formulas) is not implemented.
• IT01 to IT4 and the Δ values that depend on them are approximate (the standard's own description); enter the table value in the override field when a gauge-grade fit matters.
• Temperature is not treated: the ISO system refers to 20 °C, and a fit of dissimilar materials changes with temperature by the difference of the expansions.
• No form or orientation tolerances other than the position-at-MMC arithmetic; no gauge-design allowances (gauge-maker's tolerance and wear allowance).
• The statistical section assumes independent, centred normal processes; see the Tolerance Stack-Up tool for other assumptions.