Limits, fits and tolerances

Why it matters

A gudgeon pin must slide in its piston boss but be firm in the small end; a bearing must be pressed onto a shaft yet slide into its housing; a gearbox must accept a replacement gear from any supplier. All of this is specified with limits and fits. A correct tolerance gives function and interchangeability at the lowest cost; a needlessly tight one multiplies machining cost, and a loose one causes noise, wear or seizure.

Key ideas

Basic vocabulary (for a hole and a shaft).

• Basic (nominal) size – the size from which limits are derived (e.g. 40 mm).
• Limits of size – the maximum and minimum permissible sizes.
• Deviation – limit minus basic size. Upper deviation: ES for holes, es for shafts. Lower deviation: EI for holes, ei for shafts.
• Tolerance – maximum limit − minimum limit = upper deviation − lower deviation (always positive).
• Fundamental deviation – the deviation closest to the zero line; it fixes the tolerance zone's position and is denoted by a letter.
• Allowance – the intentional difference between the maximum material limits of mating parts: minimum clearance (positive) or maximum interference (negative).
• Unilateral tolerance lies on one side of the basic size (e.g. 40 +0.025/0); bilateral on both (40 ± 0.02).

Fits.

• Clearance fit – the smallest hole is larger than or equal to the largest shaft; there is always clearance (or zero in the limiting case). Bearings, sliding pins.
• Interference fit – the largest hole is smaller than the smallest shaft; assembly needs pressing or shrinking (heating the hub or cooling the shaft). Valve seats, gear hubs, bearing races on rotating shafts.
• Transition fit – depending on actual sizes there may be a small clearance or a small interference. Location of gears, couplings and spigots.

Hole-basis vs shaft-basis. In the hole-basis system the hole has a fixed position (H: lower deviation zero) and different shafts give different fits; it is preferred because holes are made with fixed-size tools (drills, reamers) and checked with fixed plug gauges, whereas shafts are easily turned or ground to any size. Shaft-basis (h shaft: upper deviation zero) is used when one standard shaft (e.g. bright drawn bar, a bearing outer race) must mate with several holes.

ISO system (ISO 286, adopted in India as IS 919). A fit is written as basic size + hole letter and grade / shaft letter and grade, e.g. 40 H7/g6.

• Letters give the fundamental deviation: capital for holes (A … ZC), lower-case for shafts (a … zc). H and h have zero fundamental deviation. Shafts a–g lie below the zero line (clearance with H holes), js/j/k/m/n are transition, and p–zc lie above it (interference). The pattern is the mirror image for holes.
• Grades IT01, IT0, IT1 … IT18 give the tolerance magnitude: IT5–IT7 for precision grinding and reaming, IT7–IT9 for fine turning and boring, IT11–IT14 for rough work. In a fit the hole is usually one grade coarser than the shaft (holes are harder to make accurately).
• Tolerance values come from the standard tolerance unit i for the diameter step, multiplied by a grade factor (IT6 = 10i, IT7 = 16i, IT8 = 25i, IT9 = 40i, IT10 = 64i, IT11 = 100i for IT5–IT16 in the R5 progression).
• Diameter steps (e.g. 18–30, 30–50, 50–80 mm) are used with their geometric mean.
• In practice, read deviations directly from the standard tables.

Common hole-basis fits. H7/g6 (close running/sliding), H7/h6 (locational clearance), H7/k6 (transition, accurate location), H7/n6 (tighter transition), H7/p6 (light press), H7/s6 (medium drive).

Tolerance stack-up. In an assembly chain the worst-case tolerance of the resultant dimension is the sum of the individual tolerances. The statistical (root-sum-square) method assumes random, centred, independent variation and gives a smaller, more realistic band. Selective assembly (grading parts into groups and mating like with like) achieves tight fits from wider manufacturing tolerances.

Formulas

T_h = Hmax − Hmin = ES − EI T_s = Smax − Smin = es − ei Hole and shaft tolerance (mm or µm).

C_max = Hmax − Smin C_min = Hmin − Smax Maximum and minimum clearance (mm); a negative value is an interference.

I_max = Smax − Hmin Maximum interference (mm), for interference and transition fits.

T_fit = C_max − C_min = T_h + T_s Fit tolerance (variation of clearance).

i = 0.45·D^(1/3) + 0.001·D Standard tolerance unit (µm) for sizes up to 500 mm; D = geometric mean of the diameter step, D = √(D₁·D₂) (mm).

es (g) = −2.5·D^0.34 es (f) = −5.5·D^0.41 es (e) = −11·D^0.41 es (d) = −16·D^0.44 Empirical formulas for the upper deviation of common clearance shafts (µm), as given in IS 919/ISO-based textbooks; use the standard tables for exact rounded values.

T_worst = Σ T_k T_RSS = √(Σ T_k²) Worst-case and root-sum-square stack-up of a chain of dimensions (T_k = ± half-tolerances, or full tolerances used consistently).

Worked examples

Example 1 (standard) – limits of a 40 H7/g6 fit.

1. Diameter step 30–50 mm: D = √(30 × 50) = 38.73 mm.
2. i = 0.45·D^(1/3) + 0.001·D = 0.45 × 3.383 + 0.039 = 1.561 µm.
3. IT7 = 16i = 25.0 → 25 µm; IT6 = 10i = 15.6 → 16 µm.
4. H hole: EI = 0, ES = +25 µm ⇒ hole 40.000 – 40.025 mm.
5. g shaft: es = −2.5 × 38.73^0.34 = −8.7 → −9 µm; ei = −9 − 16 = −25 µm ⇒ shaft 39.975 – 39.991 mm.
6. C_max = 40.025 − 39.975 = 0.050 mm; C_min = 40.000 − 39.991 = 0.009 mm – a clearance fit. These match the tabulated ISO values (+25/0 and −9/−25 µm).

Example 2 (GATE level) – stack-up in a slot. Three blocks of 20 ± 0.02, 30 ± 0.03 and 50 ± 0.05 mm are placed in a slot of 100.20 ± 0.04 mm. Find the gap.

1. Block stack: nominal 100.00, worst-case ± (0.02 + 0.03 + 0.05) = ± 0.10 ⇒ 99.90 to 100.10 mm.
2. Maximum gap = 100.24 − 99.90 = 0.34 mm; minimum gap = 100.16 − 100.10 = 0.06 mm (worst case: nominal 0.20 ± 0.14).
3. Statistical estimate: ±√(0.02² + 0.03² + 0.05² + 0.04²) = ±0.073 mm ⇒ gap ≈ 0.20 ± 0.073 mm, i.e. about 0.13 to 0.27 mm.

Example 3 – clearance from given deviations. Hole 40 (+0.04/−0.02), shaft 40 (+0.01/−0.03): C_max = 40.04 − 39.97 = 0.07 mm; C_min = 39.98 − 40.01 = −0.03 mm, i.e. up to 0.03 mm interference – a transition fit.

Common mistakes

• Calling the hole basis "the hole is at basic size": it means the hole's lower limit is the basic size (EI = 0).
• Confusing allowance (difference at maximum material conditions) with tolerance (variation of one part).
• Mixing µm deviations with mm sizes without converting.
• Using the actual size instead of the geometric mean of the diameter step in the i formula.
• Reading interference as positive clearance – keep the sign: C_min < 0 means interference.
• Assuming a transition fit is "half-way between" – it is defined by overlapping zones, and the result can be either clearance or interference.

For GATE ME

Expect numericals on limits, maximum/minimum clearance or interference, the i formula and IT grades, fundamental deviations for ISO fits, identification of fit type from given deviations, and worst-case stack-up. Conceptual questions cover hole-basis vs shaft-basis, letters and grades, and selective assembly. Practise one complete ISO fit calculation by hand.

Quick check

1. In the hole-basis system, which deviation of the hole is zero?
2. Hole 25.000–25.021, shaft 25.022–25.035 mm: what type of fit?
3. What is IT7 in terms of i?
4. Maximum clearance for hole 50.00–50.03 and shaft 49.95–49.98 mm?
5. Which is usually the coarser grade in a fit, hole or shaft?

Answers: 1. The lower deviation EI; 2. Interference fit (smallest shaft exceeds largest hole); 3. 16i; 4. 0.08 mm; 5. The hole.