Study guide
14.1 Limits, Fits & Geometric Dimensioning & Tolerancing (GD&T)
PE Mechanical: Machine Design and Materials Exam
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1 Chapter 1: Basic Engineering Practice: Drawings, Economics & Design Methodology
1.1 Engineering Terms, Symbols, Drawing Interpretation & Conventions 1.2 Project Management & Engineering Economics 1.3 Design Methodology, Risk Assessment & Safety
2 Chapter 2: Physical & Mechanical Properties of Engineering Materials
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3 Chapter 3: Mechanics of Materials: Fundamental Loadings & Deflections
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4 Chapter 4: Mechanics of Materials: Stability, Multiaxial Stress & Thermal Effects
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5 Chapter 5: Failure Theories: Static Yielding & Fracture Mechanics
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6 Chapter 6: Fatigue Analysis & Dynamic Failure
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7 Chapter 7: Mechanical Attachments: Fasteners & Bolted Joints
7.1 Threaded Fasteners, Proof Strength & Preload Mechanics 7.2 Bolted Joint Stiffness, External Loading & Separation 7.3 Non-Threaded Fasteners: Keys, Pins, Clevises, Retaining Rings, Rivets, Lugs & Anchors
8 Chapter 8: Mechanical Attachments: Permanent Joints & Bonding
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9 Chapter 9: Power Transmission: Shafts & Gearing
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10 Chapter 10: Power Transmission: Bearings, Flexible Drives, Friction Devices & Prime Movers
10.1 Rolling-Element Bearings & L10 Life Calculations 10.2 Hydrodynamic Journal Bearings & Lubrication Regimes 10.3 Belts, Roller Chains & Flexible Drives 10.4 Clutches, Brakes & Power Screws 10.5 Electric Motors, Engines & Flywheels
11 Chapter 11: Mechanical Components: Pressure Vessels & Structural Systems
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12 Chapter 12: Mechanical Dynamics, Vibrations & Fluid Power
12.1 Single-Degree-of-Freedom Vibrations & Resonance 12.2 Vibration Isolation, Multi-DOF Systems & Transmissibility 12.3 Fluid Power Systems: Hydraulics & Pneumatics
13 Chapter 13: Kinematics, Mechatronics & Control Systems
13.1 Mechanism Kinematics, Linkages, Cams & Simple Machines 13.2 Basic Mechatronics, Sensors & Closed-Loop Controls
14 Chapter 14: Supportive Knowledge: Fits, Tolerances & Manufacturing Methods
14.1 Limits, Fits & Geometric Dimensioning & Tolerancing (GD&T) 14.2 Manufacturing Methods, Machinability, Heat Treatment & Assembly
15 Chapter 15: Supportive Knowledge: Codes, Computation, Testing & Corrosion
15.1 Engineering Codes & Standards 15.2 Computational Methods: FEA & Numerical Analysis 15.3 Instrumentation, Testing, Inspection & Quality 15.4 Chemical Processes: Corrosion, Oxidation & Embrittlement
14.1 Limits, Fits & Geometric Dimensioning & Tolerancing (GD&T)
Key Takeaways Explain This Topic AI
The ISO/ANSI limit and fit system defines clearance, transition, and interference fits; the hole-basis system (basic hole H H H with lower deviation E I = 0 EI = 0 E I = 0 ) is standard industrial practice because standardized tooling (drills, reamers, broaches) fixes hole size while shaft diameters are easily turned or ground to fit.
The ASME Y14.5 Datum Reference Frame constrains the six spatial degrees of freedom using the 3-2-1 principle: the primary datum establishes 3 points of contact, the secondary datum establishes 2 points, and the tertiary datum establishes 1 point.
Form tolerances (flatness, straightness, circularity, cylindricity) never reference datums, whereas orientation, location, and runout tolerances require datum references to establish true geometric orientation or position.
Under Maximum Material Condition (MMC), any departure of the produced feature size from its MMC size boundary provides an equal 'bonus tolerance,' expanding the permissible position tolerance zone while preserving virtual condition assembly boundaries.
Tolerance stack-up analysis contrasts deterministic worst-case arithmetic summation ( T W C = ∑ t i T_{WC} = \sum t_i T W C = ∑ t i ) with statistical Root-Sum-Square ( T R S S = ∑ t i 2 T_{RSS} = \sqrt{\sum t_i^2} T RSS = ∑ t i 2 ), where RSS assumes independent normal distributions at 3 σ 3\sigma 3 σ process limits.
Last updated: October 2026
14.1 Limits, Fits & Geometric Dimensioning & Tolerancing (GD&T)
Precision mechanical design requires exact dimensional control to ensure interchangeability, reliable functional performance, and cost-effective manufacturing. Components fabricated in high-volume production invariably exhibit geometric variations resulting from tool wear, machine vibration, thermal expansion, and metallurgical inhomogeneities. Tolerancing bridges theoretical kinematic design and shop-floor fabrication by defining permissible dimensional and geometric boundaries.
Limits and Fits: Fundamental Terminology & ISO/ANSI Systems
Standardized fit systems classify the relative assembly relationship between two mating cylindrical features—generically designated as a hole (internal feature) and a shaft (external feature). +------------------------------------+ --- Maximum Hole Limit
- | HOLE TOLERANCE |
- Zero Line --------+------------------------------------+ --- Basic Size / Min Hole (EI=0)
- (Basic Size) | |
- | CLEARANCE |
- | |
- -----------------+------------------------------------+ --- Max Shaft Limit (es)
- | | SHAFT TOLERANCE |
- | +------------------------------------+ --- Min Shaft Limit (ei)
- | |
- +------------------------------------------------------+ --- Shaft Lower Boundary
- Core Dimensional Definitions
Basic Size ( D , d D, d D , d ) : The theoretical nominal size from which design limits of size are established by the application of allowances and tolerances.
Actual Size : The measured physical size of an individual manufactured feature.
Limits of Size : The maximum and minimum permissible dimensions for a feature ( D max , D min D_{\max}, D_{\min} D m a x , D m i n for holes; d max , d min d_{\max}, d_{\min} d m a x , d m i n for shafts).
Tolerance ( T T T ) : The total permissible variation of a feature's size: T hole = D max − D min = E S − E I T_{\text{hole}} = D_{\max} - D_{\min} = ES - EI T hole = D m a x − D m i n = ES − E I T shaft = d max − d min = e s − e i T_{\text{shaft}} = d_{\max} - d_{\min} = es - ei T shaft = d m a x − d m i n = es − e i
Deviations : The algebraic difference between a size limit and the corresponding basic size:
Upper Deviation ( E S ES ES for holes, e s es es for shafts) : E S = D max − D , e s = d max − d ES = D_{\max} - D, \quad es = d_{\max} - d ES = D m a x − D , es = d m a x − d
Lower Deviation ( E I EI E I for holes, e i ei e i for shafts) : E I = D min − D , e i = d min − d EI = D_{\min} - D, \quad ei = d_{\min} - d E I = D m i n − D , e i = d m i n − d
Fundamental Deviation : The deviation that defines the position of the tolerance zone relative to the zero line (the basic size). Under the ISO system, fundamental deviations are designated by uppercase letters ( A A A through Z C ZC ZC ) for holes and lowercase letters ( a a a through z c zc zc ) for shafts.
Standard International Tolerance (IT) Grades
Standard tolerance grades designate the magnitude of the tolerance spread across standardized dimensional steps, spanning IT01, IT0, IT1 to IT16 :
IT01 to IT4 : Precision gage making, optical apparatus, lapping, and fine honing.
IT5 to IT7 : Precision engineering, grinding, reaming, ball and roller bearing fits, precision spindle bores.
IT8 to IT11 : General mechanical engineering, commercial turning, milling, boring, and cold-drawn components.
IT12 to IT16 : Coarse manufacturing, sheet metal stamping, sand casting, flame cutting, and rough forging.
The standard tolerance unit i i i (in μ m \mu\text{m} μ m ) for basic sizes up to 500 mm 500\,\text{mm} 500 mm is computed from the geometric mean diameter D = D 1 D 2 D = \sqrt{D_1 D_2} D = D 1 D 2 of the size band: i = 0.45 D 3 + 0.001 D [ μ m ] i = 0.45 \sqrt[3]{D} + 0.001 D \quad [\mu\text{m}] i = 0.45 3 D + 0.001 D [ μ m ]
Representative IT grade multipliers relative to i i i include:
- IT5 = 7 i \text{IT5} = 7i IT5 = 7 i
- IT6 = 10 i \text{IT6} = 10i IT6 = 10 i
- IT7 = 16 i \text{IT7} = 16i IT7 = 16 i
- IT8 = 25 i \text{IT8} = 25i IT8 = 25 i
- IT9 = 40 i \text{IT9} = 40i IT9 = 40 i
- IT10 = 64 i \text{IT10} = 64i IT10 = 64 i
- IT11 = 100 i \text{IT11} = 100i IT11 = 100 i
- Fit Classifications: Clearance, Transition, and Interference
Mating fits are categorized into three primary functional regimes based on relative tolerance zone placement:
Fit Classification Defining Condition Industrial Application Preferred Examples
Clearance Fit D min > d max D_{\min} > d_{\max} D m i n > d m a x (Allowance > 0 > 0 > 0 , always positive clearance) Journal bearings, sliding guides, removable locating pins H11/c11 \text{H11/c11} H11/c11 (loose), H9/d9 \text{H9/d9} H9/d9 (free running), H8/f7 \text{H8/f7} H8/f7 (close running), H7/g6 \text{H7/g6} H7/g6 (sliding), H7/h6 \text{H7/h6} H7/h6 (locational)
Transition Fit Tolerance zones overlap; may yield clearance or interference Accurate locating dowels, rigid spigots, lightly keyed pulleys H7/k6 \text{H7/k6} H7/k6 (true locational), H7/n6 \text{H7/n6} H7/n6 (light press locating)
Interference Fit D max < d min D_{\max} < d_{\min} D m a x < d m i n (Negative clearance, always interference) Shrink fits, permanent press-fitted gears, bearing inner rings on shafts H7/p6 \text{H7/p6} H7/p6 (light drive), H7/s6 \text{H7/s6} H7/s6 (medium drive), H7/u6 \text{H7/u6} H7/u6 (heavy force/shrink)
Fit Quantities
Minimum Clearance ( C min C_{\min} C m i n ) : C min = D min − d max = E I − e s C_{\min} = D_{\min} - d_{\max} = EI - es C m i n = D m i n − d m a x = E I − es
Maximum Clearance ( C max C_{\max} C m a x ) : C max = D max − d min = E S − e i C_{\max} = D_{\max} - d_{\min} = ES - ei C m a x = D m a x − d m i n = ES − e i
Maximum Interference ( I max I_{\max} I m a x ) : I max = d max − D min = e s − E I I_{\max} = d_{\max} - D_{\min} = es - EI I m a x = d m a x − D m i n = es − E I
Minimum Interference ( I min I_{\min} I m i n ) : I min = d min − D max = e i − E S I_{\min} = d_{\min} - D_{\max} = ei - ES I m i n = d m i n − D m a x = e i − ES
Hole-Basis vs. Shaft-Basis System
Hole-Basis System (Basic Hole H H H ) :
The basic size is assigned to the minimum hole size ( D min = D D_{\min} = D D m i n = D ).
The lower deviation of the hole is identically zero ( E I = 0 EI = 0 E I = 0 ).
The various fits are generated by varying the shaft tolerance zone position ( c , d , f , g , h , k , n , p , s , u c, d, f, g, h, k, n, p, s, u c , d , f , g , h , k , n , p , s , u ).
Universal Industrial Preference : Standard hole-making tools (drills, reamers, broaches, plug gages) have fixed physical dimensions and are costly to customize. It is vastly more economical to purchase standardized cutting tools to produce an H 7 H7 H 7 or H 8 H8 H 8 bore and adjust the exterior shaft diameter to the desired fit on a CNC lathe or cylindrical grinder.
Shaft-Basis System (Basic Shaft h h h ) :
The basic size is assigned to the maximum shaft size ( d max = d d_{\max} = d d m a x = d ).
The upper deviation of the shaft is identically zero ( e s = 0 es = 0 es = 0 ).
Selected primarily when a single continuous, precision cold-drawn or ground steel shaft passes through multiple components requiring different fits (e.g., a clearance sleeve bearing, a keyed transition hub, and a press-fit bearing inner ring along one uninterrupted bar).
ASME Y14.5 Geometric Dimensioning & Tolerancing (GD&T)
Traditional coordinate (plus-minus) tolerancing exhibits fundamental engineering shortcomings: it produces rectangular or square tolerance zones (yielding 41.4 % 41.4\% 41.4% greater tolerance along diagonals than across axes), lacks clear functional datum precedence, and fails to communicate how parts mate in three-dimensional space.
ASME Y14.5 establishes a symbolic engineering language that explicitly communicates functional design intent, controls geometric relationships, and defines inspection requirements without ambiguity.
The Datum Reference Frame (DRF) & The 3-2-1 Principle
An unconstrained rigid body possesses six spatial degrees of freedom (DOF) : three mutually perpendicular translations ( T x , T y , T z T_x, T_y, T_z T x , T y , T z ) and three rotations ( R x , R y , R z R_x, R_y, R_z R x , R y , R z ). The Datum Reference Frame constrains these DOFs using the 3-2-1 contact principle : Z ^ (Normal to Primary)
- |
- | +-----------------------+
- | / /|
- | / SECONDARY DATUM [B] / |
- |/ (2 contact points)/ |
- [C] TERTIARY DATUM ------------+-----------------------+ |
- (1 contact point) | | |
- | PRIMARY DATUM [A] | + ---> Y
- | (3 contact points) | /
- | | /
- +-----------------------+/
- /
- /
- v X
Primary Datum (Datum A) : Contacts the part at a minimum of three non-collinear points . Constrains 3 degrees of freedom : translation along the Z Z Z -axis ( T z T_z T z ) and rotation about the X X X and Y Y Y axes ( R x , R y R_x, R_y R x , R y ).
Secondary Datum (Datum B) : Oriented perpendicular to the primary datum, contacting the part at a minimum of two points . Constrains 2 degrees of freedom : translation along the Y Y Y -axis ( T y T_y T y ) and rotation about the Z Z Z -axis ( R z R_z R z ).
Tertiary Datum (Datum C) : Oriented perpendicular to both primary and secondary datums, contacting the part at a minimum of one point . Constrains the final remaining 1 degree of freedom : translation along the X X X -axis ( T x T_x T x ).
Feature Control Frame Syntax & The Geometric Characteristics
The Feature Control Frame (FCF) is a rectangular compartmented block containing all geometric tolerance instructions for a feature: +-------+-------------------------+---------+---------+---------+
- | [+] | Ø 0.25 [M] | A | B [M] | C |
- +-------+-------------------------+---------+---------+---------+
- Symbol Tol Zone & Modifiers Primary Secondary Tertiary
- (Dia 0.25 at MMC) Datum Datum Datum
The 2009 edition of ASME Y14.5 listed 14 geometric characteristics in five families. ASME Y14.5-2018 deleted concentricity and symmetry , leaving 12; the two still appear on older drawings and in ISO 1101 (the rows are kept below for recognition):
- Family Geometric Characteristic Symbol Datums Permitted? Tolerance Zone Shape
- Form Flatness ⏥ Never Two parallel planes
- Straightness — Never Two parallel lines, or cylindrical zone
- Circularity (Roundness) ○ Never Two concentric coplanar circles
- Cylindricity ⌭ Never Two coaxial cylinders
- Profile Profile of a Line ⌒ Optional Two non-uniform offset lines
- Profile of a Surface ⌓ Optional / Recommended Two offset boundary surfaces
- Orientation Perpendicularity ⟂ Mandatory Two parallel planes, or cylindrical zone
- Parallelism ∥ Mandatory Two parallel planes, or cylindrical zone
- Angularity ∠ Mandatory Two parallel planes inclined at basic angle
- Location Position ⌖ Mandatory Cylindrical zone ( ∅ \varnothing ∅ ) or two parallel planes
- Concentricity ◎ Mandatory Cylindrical zone centered on datum axis (deleted in Y14.5-2018)
- Symmetry ⌯ Mandatory Two parallel planes centered on datum plane (deleted in Y14.5-2018)
- Runout Circular Runout ↗ Mandatory (Axis) Two coplanar circles during 360 ∘ 360^\circ 36 0 ∘ rotation
- Total Runout ⌰ Mandatory (Axis) Two coaxial cylinders / parallel planes during rotation + axial traverse
Important
Form tolerances never reference a datum. Flatness, straightness, circularity, and cylindricity evaluate a feature solely against its own ideal geometric form, independent of any external coordinate system or datum plane.
Material Condition Modifiers & Bonus Tolerance Mechanics
Material condition modifiers determine how geometric tolerances interact with feature size variations:
Maximum Material Condition (MMC, M ◯ \textcircled{\tiny M} M ◯ ) :
The condition in which a feature of size contains the maximum amount of material within its stated dimensional limits.
Internal feature (hole) : Minimum hole diameter ( D min D_{\min} D m i n ).
External feature (shaft/pin) : Maximum pin diameter ( d max d_{\max} d m a x ).
Least Material Condition (LMC, L ◯ \textcircled{\tiny L} L ◯ ) :
The condition in which a feature of size contains the minimum amount of material.
Internal feature (hole) : Maximum hole diameter ( D max D_{\max} D m a x ).
External feature (shaft/pin) : Minimum pin diameter ( d min d_{\min} d m i n ).
Used to protect minimum wall thickness and prevent structural breakout.
Regardless of Feature Size (RFS) :
The stated geometric tolerance applies strictly at all produced feature sizes.
Default rule under ASME Y14.5 (the old S ◯ \textcircled{\tiny S} S ◯ symbol is obsolete).
Virtual Condition ( V C VC V C )
Virtual condition is the worst-case boundary generated by the collective effect of a feature's size limit at MMC and its stated geometric tolerance:
For an Internal Feature (Hole) : V C = MMC Size − Stated Geometric Tolerance = D min − t pos VC = \text{MMC Size} - \text{Stated Geometric Tolerance} = D_{\min} - t_{\text{pos}} V C = MMC Size − Stated Geometric Tolerance = D m i n − t pos
For an External Feature (Shaft/Pin) : V C = MMC Size + Stated Geometric Tolerance = d max + t pos VC = \text{MMC Size} + \text{Stated Geometric Tolerance} = d_{\max} + t_{\text{pos}} V C = MMC Size + Stated Geometric Tolerance = d m a x + t pos
Bonus Tolerance Derivation
When a feature control frame specifies MMC ( M ◯ \textcircled{\tiny M} M ◯ ), the stated tolerance t stated t_{\text{stated}} t stated applies strictly when the produced feature is at its MMC size. If the feature departs from MMC toward LMC, the difference between the actual produced size and the MMC size is added directly to the permissible geometric tolerance as bonus tolerance : Bonus Tolerance = ∣ Actual Feature Size − MMC Size ∣ \text{Bonus Tolerance} = |\text{Actual Feature Size} - \text{MMC Size}| Bonus Tolerance = ∣ Actual Feature Size − MMC Size ∣ Total Permissible Tolerance = t stated + Bonus Tolerance \text{Total Permissible Tolerance} = t_{\text{stated}} + \text{Bonus Tolerance} Total Permissible Tolerance = t stated + Bonus Tolerance Feature Size: MMC (Min Hole D_min) -----------------> LMC (Max Hole D_max)
- Permissible Tol: Stated Tol (t_0) -----------------> t_0 + (D_max - D_min)
- |<- Base Tolerance ->|<- Full Available Bonus Tolerance ->|
This principle significantly increases manufacturing yields by accepting parts whose position deviates further than the base tolerance, provided their larger size still ensures clearance and prevents functional assembly interference.
Tolerance Stack-Up Analysis: Worst-Case vs. Statistical RSS
Tolerance stack-up analysis calculates the cumulative variation of an assembly gap or critical functional dimension resulting from individual component tolerances along a one-dimensional kinematic loop. +------------- A1 ------------+------- A2 -------+------- A3 -------+
- | |
- +--------------------------- TOTAL LENGTH L ------------------------+
- | |<-GAP-> |
- +------------------------------ B1 ------------------------+--------+
- 1. Worst-Case (Deterministic / Arithmetic) Stack-Up
Worst-case analysis assumes that every component in the assembly chain is manufactured concurrently at its extreme worst-case tolerance limit in the direction that minimizes (or maximizes) the gap: T W C = ∑ i = 1 n t i T_{WC} = \sum_{i=1}^n t_i T W C = i = 1 ∑ n t i Gap min = ∑ A i , min − ∑ B j , max \text{Gap}_{\min} = \sum A_{i,\min} - \sum B_{j,\max} Gap m i n = ∑ A i , m i n − ∑ B j , m a x Gap max = ∑ A i , max − ∑ B j , min \text{Gap}_{\max} = \sum A_{i,\max} - \sum B_{j,\min} Gap m a x = ∑ A i , m a x − ∑ B j , m i n
Characteristics : Guarantees 100 % 100\% 100% assembly interchangeability with zero non-conformance. However, it forces excessively tight individual part tolerances, drastically driving up machining, tooling, and scrap costs.
2. Statistical Root-Sum-Square (RSS) Stack-Up
Statistical tolerancing is rooted in the Central Limit Theorem. If individual component dimensions vary independently and follow normal (Gaussian) distributions centered between their limits, the probability of all parts simultaneously hitting their worst-case extremes is vanishingly small. The assembly variance equals the sum of the individual variances: σ assembly 2 = ∑ i = 1 n σ i 2 \sigma_{\text{assembly}}^2 = \sum_{i=1}^n \sigma_i^2 σ assembly 2 = i = 1 ∑ n σ i 2
Assuming each part tolerance t i t_i t i represents a 3 σ 3\sigma 3 σ process boundary ( t i = 3 σ i t_i = 3\sigma_i t i = 3 σ i ): T R S S = 3 σ assembly = 3 ∑ i = 1 n ( t i 3 ) 2 = ∑ i = 1 n t i 2 T_{RSS} = 3\sigma_{\text{assembly}} = 3\sqrt{\sum_{i=1}^n \left( \frac{t_i}{3} \right)^2} = \sqrt{\sum_{i=1}^n t_i^2} T RSS = 3 σ assembly = 3 i = 1 ∑ n ( 3 t i ) 2 = i = 1 ∑ n t i 2 Nominal Gap = μ gap = ∑ A i , nom − ∑ B j , nom \text{Nominal Gap} = \mu_{\text{gap}} = \sum A_{i,\text{nom}} - \sum B_{j,\text{nom}} Nominal Gap = μ gap = ∑ A i , nom − ∑ B j , nom Gap RSS limits = μ gap ± T R S S \text{Gap}_{\text{RSS limits}} = \mu_{\text{gap}} \pm T_{RSS} Gap RSS limits = μ gap ± T RSS
Modified RSS / Bender Factor : In real-world manufacturing, processes may suffer mean shifts or tool wear drifts. A correction factor C f C_f C f (typically 1.3 1.3 1.3 to 1.5 1.5 1.5 ) is applied to standard RSS: T mod RSS = C f ∑ i = 1 n t i 2 T_{\text{mod RSS}} = C_f \sqrt{\sum_{i=1}^n t_i^2} T mod RSS = C f i = 1 ∑ n t i 2
- Worked Engineering Examples
- Worked Example 1: 1D Tolerance Stack-Up Comparison (Worst-Case vs. RSS)
- Problem Statement
A high-speed pump shaft assembly contains three precision spacers ( A 1 , A 2 , A 3 A_1, A_2, A_3 A 1 , A 2 , A 3 ) stacked axially inside a housing bore ( B B B ). An axial clearance gap G G G between the end of spacer A 3 A_3 A 3 and the housing shoulder must be maintained to prevent bearing preloading upon thermal expansion.
Component dimensional specifications:
- Housing bore depth ( B B B ): 75.00 ± 0.12 mm 75.00 \pm 0.12\,\text{mm} 75.00 ± 0.12 mm
- Spacer 1 length ( A 1 A_1 A 1 ): 25.00 ± 0.04 mm 25.00 \pm 0.04\,\text{mm} 25.00 ± 0.04 mm
- Spacer 2 length ( A 2 A_2 A 2 ): 30.00 ± 0.05 mm 30.00 \pm 0.05\,\text{mm} 30.00 ± 0.05 mm
- Spacer 3 length ( A 3 A_3 A 3 ): 19.50 ± 0.04 mm 19.50 \pm 0.04\,\text{mm} 19.50 ± 0.04 mm
Determine:
The nominal axial clearance gap G nom G_{\text{nom}} G nom .
The worst-case minimum and maximum clearance gaps ( G W C , min , G W C , max G_{WC,\min}, G_{WC,\max} G W C , m i n , G W C , m a x ) and total worst-case tolerance T W C T_{WC} T W C .
The statistical RSS tolerance T R S S T_{RSS} T RSS and the predicted 3 σ 3\sigma 3 σ clearance limits ( G R S S , min , G R S S , max G_{RSS,\min}, G_{RSS,\max} G RSS , m i n , G RSS , m a x ).
Step-by-Step Solution Step 1: Nominal Gap Calculation G nom = B nom − ( A 1 , nom + A 2 , nom + A 3 , nom ) G_{\text{nom}} = B_{\text{nom}} - (A_{1,\text{nom}} + A_{2,\text{nom}} + A_{3,\text{nom}}) G nom = B nom − ( A 1 , nom + A 2 , nom + A 3 , nom ) G nom = 75.00 − ( 25.00 + 30.00 + 19.50 ) = 75.00 − 74.50 = 0.50 mm G_{\text{nom}} = 75.00 - (25.00 + 30.00 + 19.50) = 75.00 - 74.50 = 0.50\,\text{mm} G nom = 75.00 − ( 25.00 + 30.00 + 19.50 ) = 75.00 − 74.50 = 0.50 mm Step 2: Worst-Case Tolerance Stack-Up
Identify individual bilateral half-tolerances t i t_i t i :
- t B = 0.12 mm t_B = 0.12\,\text{mm} t B = 0.12 mm
- t A 1 = 0.04 mm t_{A1} = 0.04\,\text{mm} t A 1 = 0.04 mm
- t A 2 = 0.05 mm t_{A2} = 0.05\,\text{mm} t A 2 = 0.05 mm
- t A 3 = 0.04 mm t_{A3} = 0.04\,\text{mm} t A 3 = 0.04 mm
Compute total worst-case tolerance: T W C = t B + t A 1 + t A 2 + t A 3 = 0.12 + 0.04 + 0.05 + 0.04 = 0.25 mm T_{WC} = t_B + t_{A1} + t_{A2} + t_{A3} = 0.12 + 0.04 + 0.05 + 0.04 = 0.25\,\text{mm} T W C = t B + t A 1 + t A 2 + t A 3 = 0.12 + 0.04 + 0.05 + 0.04 = 0.25 mm
Compute worst-case gap limits: G W C , min = B min − ( A 1 , max + A 2 , max + A 3 , max ) = ( 75.00 − 0.12 ) − ( 25.04 + 30.05 + 19.54 ) G_{WC,\min} = B_{\min} - (A_{1,\max} + A_{2,\max} + A_{3,\max}) = (75.00 - 0.12) - (25.04 + 30.05 + 19.54) G W C , m i n = B m i n − ( A 1 , m a x + A 2 , m a x + A 3 , m a x ) = ( 75.00 − 0.12 ) − ( 25.04 + 30.05 + 19.54 ) G W C , min = 74.88 − 74.63 = 0.25 mm G_{WC,\min} = 74.88 - 74.63 = 0.25\,\text{mm} G W C , m i n = 74.88 − 74.63 = 0.25 mm G W C , max = B max − ( A 1 , min + A 2 , min + A 3 , min ) = ( 75.00 + 0.12 ) − ( 24.96 + 29.95 + 19.46 ) G_{WC,\max} = B_{\max} - (A_{1,\min} + A_{2,\min} + A_{3,\min}) = (75.00 + 0.12) - (24.96 + 29.95 + 19.46) G W C , m a x = B m a x − ( A 1 , m i n + A 2 , m i n + A 3 , m i n ) = ( 75.00 + 0.12 ) − ( 24.96 + 29.95 + 19.46 ) G W C , max = 75.12 − 74.37 = 0.75 mm G_{WC,\max} = 75.12 - 74.37 = 0.75\,\text{mm} G W C , m a x = 75.12 − 74.37 = 0.75 mm Gap W C = 0.50 ± 0.25 mm [ 0.25 mm to 0.75 mm ] \text{Gap}_{WC} = 0.50 \pm 0.25\,\text{mm} \quad [0.25\,\text{mm} \text{ to } 0.75\,\text{mm}] Gap W C = 0.50 ± 0.25 mm [ 0.25 mm to 0.75 mm ] Step 3: Statistical RSS Tolerance Stack-Up
Compute RSS tolerance: T R S S = t B 2 + t A 1 2 + t A 2 2 + t A 3 2 = ( 0.12 ) 2 + ( 0.04 ) 2 + ( 0.05 ) 2 + ( 0.04 ) 2 T_{RSS} = \sqrt{t_B^2 + t_{A1}^2 + t_{A2}^2 + t_{A3}^2} = \sqrt{(0.12)^2 + (0.04)^2 + (0.05)^2 + (0.04)^2} T RSS = t B 2 + t A 1 2 + t A 2 2 + t A 3 2 = ( 0.12 ) 2 + ( 0.04 ) 2 + ( 0.05 ) 2 + ( 0.04 ) 2 T R S S = 0.0144 + 0.0016 + 0.0025 + 0.0016 = 0.0201 = 0.1418 mm ≈ 0.14 mm T_{RSS} = \sqrt{0.0144 + 0.0016 + 0.0025 + 0.0016} = \sqrt{0.0201} = 0.1418\,\text{mm} \approx 0.14\,\text{mm} T RSS = 0.0144 + 0.0016 + 0.0025 + 0.0016 = 0.0201 = 0.1418 mm ≈ 0.14 mm
Compute statistical 3 σ 3\sigma 3 σ gap limits: G R S S , min = G nom − T R S S = 0.50 − 0.1418 = 0.3582 mm ≈ 0.36 mm G_{RSS,\min} = G_{\text{nom}} - T_{RSS} = 0.50 - 0.1418 = 0.3582\,\text{mm} \approx 0.36\,\text{mm} G RSS , m i n = G nom − T RSS = 0.50 − 0.1418 = 0.3582 mm ≈ 0.36 mm G R S S , max = G nom + T R S S = 0.50 + 0.1418 = 0.6418 mm ≈ 0.64 mm G_{RSS,\max} = G_{\text{nom}} + T_{RSS} = 0.50 + 0.1418 = 0.6418\,\text{mm} \approx 0.64\,\text{mm} G RSS , m a x = G nom + T RSS = 0.50 + 0.1418 = 0.6418 mm ≈ 0.64 mm Gap R S S = 0.50 ± 0.14 mm [ 0.36 mm to 0.64 mm ] \text{Gap}_{RSS} = 0.50 \pm 0.14\,\text{mm} \quad [0.36\,\text{mm} \text{ to } 0.64\,\text{mm}] Gap RSS = 0.50 ± 0.14 mm [ 0.36 mm to 0.64 mm ]
Note
The statistical RSS tolerance spread ( 0.14 mm 0.14\,\text{mm} 0.14 mm ) is 43 % 43\% 43% narrower than the deterministic worst-case spread ( 0.25 mm 0.25\,\text{mm} 0.25 mm ). This demonstrates that individual part tolerances can be significantly relaxed under statistical quality control without risking assembly interference.
- Worked Example 2: GD&T True Position, MMC Bonus Tolerance & Verification
- Problem Statement
A structural mounting bracket contains four clearance holes for M10 bolts. The engineering drawing specifies the following feature control frame and size limits for each hole:
- Stated size: ∅ 10.50 ± 0.20 mm \varnothing 10.50 \pm 0.20\,\text{mm} ∅ 10.50 ± 0.20 mm
- Geometric tolerance: [⌖ | ⌀ 0.30 Ⓜ | A | B | C]
Basic true position coordinates: X = 60.00 mm X = 60.00\,\text{mm} X = 60.00 mm , Y = 40.00 mm Y = 40.00\,\text{mm} Y = 40.00 mm
During coordinate measuring machine (CMM) inspection, Hole #1 is evaluated:
Measured diameter: D act = 10.62 mm D_{\text{act}} = 10.62\,\text{mm} D act = 10.62 mm
Measured center coordinates: X act = 60.10 mm X_{\text{act}} = 60.10\,\text{mm} X act = 60.10 mm , Y act = 40.12 mm Y_{\text{act}} = 40.12\,\text{mm} Y act = 40.12 mm
Determine:
The Maximum Material Condition (MMC) size and Least Material Condition (LMC) size.
The virtual condition boundary of the hole.
The actual radial and diametral position deviation of the produced hole center.
The bonus tolerance and total allowable position tolerance at the produced size.
Whether Hole #1 conforms to the GD&T specification.
Step-by-Step Solution Step 1: Material Condition Sizes
For an internal feature (hole): MMC Size = D min = 10.50 − 0.20 = 10.30 mm \text{MMC Size} = D_{\min} = 10.50 - 0.20 = 10.30\,\text{mm} MMC Size = D m i n = 10.50 − 0.20 = 10.30 mm LMC Size = D max = 10.50 + 0.20 = 10.70 mm \text{LMC Size} = D_{\max} = 10.50 + 0.20 = 10.70\,\text{mm} LMC Size = D m a x = 10.50 + 0.20 = 10.70 mm Step 2: Virtual Condition Boundary
For an internal hole at MMC: V C = D min − t stated = 10.30 mm − 0.30 mm = 10.00 mm VC = D_{\min} - t_{\text{stated}} = 10.30\,\text{mm} - 0.30\,\text{mm} = 10.00\,\text{mm} V C = D m i n − t stated = 10.30 mm − 0.30 mm = 10.00 mm
Any mating pin of diameter ≤ 10.00 mm \le 10.00\,\text{mm} ≤ 10.00 mm positioned at true position will assemble without interference. Step 3: Measured Position Deviation
Compute coordinate offsets from basic true position: Δ X = X act − X basic = 60.10 − 60.00 = + 0.10 mm \Delta X = X_{\text{act}} - X_{\text{basic}} = 60.10 - 60.00 = +0.10\,\text{mm} Δ X = X act − X basic = 60.10 − 60.00 = + 0.10 mm Δ Y = Y act − Y basic = 40.12 − 40.00 = + 0.12 mm \Delta Y = Y_{\text{act}} - Y_{\text{basic}} = 40.12 - 40.00 = +0.12\,\text{mm} Δ Y = Y act − Y basic = 40.12 − 40.00 = + 0.12 mm
Compute radial position error ( R dev R_{\text{dev}} R dev ): R dev = ( Δ X ) 2 + ( Δ Y ) 2 = ( 0.10 ) 2 + ( 0.12 ) 2 = 0.0100 + 0.0144 = 0.0244 = 0.1562 mm R_{\text{dev}} = \sqrt{(\Delta X)^2 + (\Delta Y)^2} = \sqrt{(0.10)^2 + (0.12)^2} = \sqrt{0.0100 + 0.0144} = \sqrt{0.0244} = 0.1562\,\text{mm} R dev = ( Δ X ) 2 + ( Δ Y ) 2 = ( 0.10 ) 2 + ( 0.12 ) 2 = 0.0100 + 0.0144 = 0.0244 = 0.1562 mm
Compute actual diametral position zone ( t act t_{\text{act}} t act ) consumed by the hole: t act = 2 R dev = 2 × 0.1562 mm = 0.3124 mm ≈ 0.312 mm t_{\text{act}} = 2 R_{\text{dev}} = 2 \times 0.1562\,\text{mm} = 0.3124\,\text{mm} \approx 0.312\,\text{mm} t act = 2 R dev = 2 × 0.1562 mm = 0.3124 mm ≈ 0.312 mm Step 4: Bonus Tolerance & Total Allowable Tolerance
The produced hole size is D act = 10.62 mm D_{\text{act}} = 10.62\,\text{mm} D act = 10.62 mm . Because D act > D MMC D_{\text{act}} > D_{\text{MMC}} D act > D MMC ( 10.62 > 10.30 10.62 > 10.30 10.62 > 10.30 ): Bonus Tolerance = D act − MMC Size = 10.62 − 10.30 = 0.32 mm \text{Bonus Tolerance} = D_{\text{act}} - \text{MMC Size} = 10.62 - 10.30 = 0.32\,\text{mm} Bonus Tolerance = D act − MMC Size = 10.62 − 10.30 = 0.32 mm
Compute total permissible position tolerance zone diameter: t total = t stated + Bonus = 0.30 mm + 0.32 mm = 0.62 mm t_{\text{total}} = t_{\text{stated}} + \text{Bonus} = 0.30\,\text{mm} + 0.32\,\text{mm} = 0.62\,\text{mm} t total = t stated + Bonus = 0.30 mm + 0.32 mm = 0.62 mm Step 5: Conformance Evaluation
Compare actual diametral position deviation against the total allowable limit: t act = 0.312 mm ≤ t total = 0.620 mm t_{\text{act}} = 0.312\,\text{mm} \le t_{\text{total}} = 0.620\,\text{mm} t act = 0.312 mm ≤ t total = 0.620 mm
Under RFS rules, the hole would have been rejected ( 0.312 > 0.300 0.312 > 0.300 0.312 > 0.300 ). However, because the MMC modifier was specified and the hole was machined larger than its minimum limit, the bonus tolerance expanded the permissible zone to ∅ 0.62 mm \varnothing 0.62\,\text{mm} ∅ 0.62 mm . Therefore, Hole #1 is fully conforming and accepted .