Chapter2
Limit and Fits
2.1 Terms of Sizes
Hole: designates all internal features of a part, including parts which are not
cylindrical as shown in Fig. 2-1(a).
Shaft: designates all external features of a part, including parts which are not
cylindrical as shown in Fig. 2-1(b) and (c).
(a)hole
(b)shaft
Figure 2-1 Hole and Shaft
(c)shaft
Basic Size/Nominal Size (D, d): The theoretical dimension from which the hole/shaft
limits are derived. It is given during the process of design. As shown in Fig.2-2, the
basic size of shaft is 50, i.e., d  50 . The basic size is the same for both members
of a fit, i.e., D  d . For example， D  d  50
Figure 2-2 A Shaft
Limits of Size: The applicable maximum and minimum sizes.
• Maximum limit of size (Dmax, dmax): the greater of the two limits of size. As
shown in Fig.2-2, the maximum limit of size for the shaft is 49.975 , i.e.,
dmax  49.975 .
•
Minimum limit of size (Dmin, dmin): the smaller of the two limits of size. As
shown in Fig.2-2, the minimum limit of size for the shaft is 49.950 , i.e.,
d min   49.950 .
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Actual Size (Da, da): A measure size obtained from a finished part as shown in
Fig.2-3. For example, Da  49.972 .
Fig.2-3 Actual Size
Any actual size shall not exceed the maximum limit of size or the minimum limit of
size, i.e., Dmin  Da  Dmax , d min  d a  d max , otherwise, the component is
unqualified.
Maximum Material Condition (MMC) and Maximum Material Size (MMS): the
maximum material condition is a feature where contains the maximum amount of
material within the stated limits of size. Maximum material size (DM, dM) is the limits
of size in this state, i.e., the minimum hole size Dmin and the maximum shaft size dmax
(shown in Fig.2-4). For another example, as shown in Fig.2-2, the maximum material
size for the shaft is 49.975 , i.e., d M  d max  49.975 .
Least Material Condition (LMC) and Least Material Size (LMS): the least
material condition is a feature where contains the least amount of material within the
stated limits of size. Least material size (DL, dL) is the limits of size in this state, i.e.,
the maximum hole size Dmax and the minimum shaft size dmin (shown in Fig.2-4). For
another example, as shown in Fig.2-2, the mniimum material size for the shaft
is 49.950 , i.e., d L  d min  49.950 .
Figure2-4 The Maximum Material Condition and the Least Material Condition
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2.2 Terms of Deviations and Tolerances
Deviation of Size: the algebraic difference between the limits of size and the basic
size.
 The Upper Deviation (ES, es): the algebraic difference between the maximum
size and the basic size.
 The Lower Deviation (EI, ei): the algebraic difference between the minimum
size and the basic size.
ES  Dmax  D
EI  Dmin  D
Shaft : es  d max  d
Hole :
(2-1)
ei  d min  d
Tolerance (TD, Td): The total permissible deviation of a size. It is also equal to the
difference between the limits of size. The value of tolerance is always positive.
Hole : TD  Dmax  Dmin  ES  EI
Shaft : Td  d max  d min  es  ei
(2-2)
Example1: A hole 5000.025 is known. Calculate these sizes D, Dmax, Dmin, DL, DM,
ES, EI, and TD.
Solution:
D=  50mm
Dmax=  50.025mm
Dmin=  50mm
DL= Dmax =  50.025mm
DM = Dmin =  50mm
ES=+0.025mm
EI=0
TD= Dmax -Dmin =ES- EI=0.025mm
Size Tolerance Zones: is the tolerance range of the size as shown in Fig.2-5. In the
tolerance zone diagram, an area bounded by the two lines represents the upper and
lower deviation. The tolerance zone consists of “the size of the tolerance zone” and
“the position of the tolerance zone”, the former is determined by the standard
tolerance, and the later is determined by the basic deviation.
Zero Line: In the tolerance zone diagram, the datum line used to determine the
deviation of size is called zero line as shown in Fig.2-5. Generally the zero line is
9
used to represent the basic size. It is defined that the value of deviation above the zero
line is positive, while the value of deviation below the zero line is negative.
Fig.2-5 shows a size tolerance zone where the basic size is  50, the upper deviation
is +0.008, the lower deviation is -0.008, the tolerance is 0.016.
Figure 2-5 Size Tolerance Zone
Basic Deviation: It is the upper or lower deviation used to determine the relative
position between the tolerance zone and zero line. Generally, the deviation nearer to
the zero line is treated as the basic deviation.
Example2:
As shown in Fig.2-6, the deviation of hole and shaft is Hole: 5000.025 and Shaft:
5000..025
050 . Give the graphical representation of hole and shaft.
Figure 2-6 An Example of Size Tolerance Zone
2.3 Terms of Fit
Since even the simplest machine involves the fitting together of several parts for the
purpose of design and production, it is necessary to know how the various parts fit
together. A fit between two parts to be assembled can be defined as the difference
between their sizes before assembly. Or in other words, FIT is the general term to
signify the range of tightness or looseness resulting from the application of a specific
combination of allowances and tolerances in the design of the mating parts.
Allowance: It is the dimensional difference between the maximum mating limits of
mating parts, intentionally provided to obtain the desired degree or class of fit. If the
allowance is positive, it will result in the minimum clearance between the mating
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parts, and of the allowance is negative, it will result in the maximum interference.
Clearance (X): in a fit, it is the difference between the sizes of the hole and the shaft,
before assembly. Minimum clearance is a clearance fit and is the difference between
the minimum size of the hole and the maximum size of the shaft. Maximum clearance
is the difference between the maximum size of the hole and the minimum size of the
shaft.
Interference (Y): in a fit, it is the difference between the sizes of the hole and the
shaft, before assembly. The minimum interference is the arithmetical difference
between the maximum size of the hole and the minimum size of the shaft before
assembly. The maximum interference is the arithmetical difference between the
minimum size of the hole and the maximum size of the shaft before assembly.
Fit:the relationship between the hole tolerance zone and shaft tolerance zone with the
same basic size. Fits are of three general types: clearance, interference, and transition,
depending on the actual limits of the hole or shaft. Fig. 2-7 illustrates the three types
of fits.
Figure 2-7 Three Types of Fits
Clearance Fits (shown in Fig.2-8): the difference between the hole and shaft sizes
before assembly is positive. Clearance fits have limits of size prescribed such that a
clearance always results when the mating parts are assembled. Clearance fits are
intended for the accurate assembly of parts and bearings. The parts can be assembled
by hand because the hole is always larger than the shaft. Some application examples
are shown in Fig.2-9.
X max  Dmax  d min  ES  ei
X min  Dmin  d max  EI  es
(2-3)
11
Figure 2-8 Clearance Fit
(a)
(b)
Figure 2-9 Application Examples of Clearance Fit
Transition Fits (shown in Fig.2-10): this fit may provide either clearance or
interference, depending on the actual value of the tolerance of individual parts.
Transition fits are a compromise between the clearance and interference fits. They are
used for applications where accurate location is important, but either a small amount
of clearance or interference is permissible. Some application examples are shown in
Fig.2-11.
Ymax  Dmin  d max  EI  es
X max  Dmax  d min  ES  ei
(2-4)
12
Figure 2-10 Transition Fit
(a)
(b)
Figure 2-11 Application Examples of Transition Fit
Interference Fits (shown in Fig.2-12): the arithmetic difference between the hole
and shaft sizes before assembly is negative. Interference fits have a limit of size
prescribed that an interference always results when mating parts are assembled. The
hole is always smaller than the shaft. Interference fits are for the permanent
assemblies of parts which require rigidity and alignment, such as dowel pins and
bearings in casting. Some application examples are shown in Fig.2-13.
Ymin  Dmax  d min  ES  ei
Ymax  Dmin  d max  EI  es
(2-5)
13
Figure 2-12 Interference Fit
(a)
(b)
Figure 2-13 Application Examples of Interference Fit
Fit Tolerance: allow change amount of the clearance or interference and can be
calculated as
Clearance fit:
Tf =
Interference fit: Tf =
Transition Fit:
Tf =
|Xmax - Xmin|
|Ymin - Ymax|
|Xma x - Ymax|
(2-6)
Fit Tolerance Zone: is used to represent the fit between hole and shafts as shown in
Fig.2-14.
14
Figure 2-14 Drawing of Fit Tolerance Zone
Example3:
0.025
Graphical represent three fits: hole: 50 00.025 and shaft: 50 00..025
041 , hole  50 0
and
0.025
shaft 50 00..059
and shaft 50 00..018
043 , and hole  50 0
002 , and calculate their limit
clearances or interferences.
Solution:
(1)Xmax=ES-ei=+0.025-(-0.041)=+0.066mm
Xmin=EI-es=0-(-0.025)=+0.025mm
Tf=|Xmax-Xmin|=|+0.066-(+0.025)|=0.041mm
(2)Ymax=EI-es=0-(+0.059)=-0.059mm
Ymin=ES-ei=+0.025-(+0.043)=-0.018mm
Tf=|Ymin-Ymax|=|-0.018-(-0.059)|=0.041mm
(3)Xmax=ES-ei=+0.025-(+0.002)=+0.023mm
Ymax=EI-es=0-(+0.018)=-0.018mm
Tf=|Xmax-Ymax|=|+0.023-(-0.018)|=0.041mm
The drawing of size tolerance zone is given in Fig.2-15.
Figure 2-15 Drawing of Size Tolerance Zone
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Example4:
Complete the Table2-1 for sizes of hole and shaft.
Table 2-1 Tolerances and Fits of Holes and Shafts
Basic
size
ES
EI
Φ50
es
ei
0
-0.016
Φ60
Xmax(Ymax)
+0.011
Xmin(Ymin)
TD
-0.020
0.025
0.019
Td
Tf
0.030
0.49
Solution:
(1)Ymin=ES-ei
ES= Ymin+ei=-0.020-0.016=-0.036
TD=ES-EI
EI=ES-TD=-0.036-0.025=-0.061
Td=es-ei=0.016
Ymax=EI-es=-0.061-0=-0.061
Tf= TD +Td=0.025+0.016=0.041
(2)
Td=Tf-TD=0.049-0.030=0.019
Td=es-ei, es=Td+ei=0.019+0.011=+0.030
Xmax=ES-ei
ES=Xmax+ei=0.019+0.011=+0.030
EI=ES-TD=+0.030-0.030=0
Ymax=EI-es=0-0.030=-0.030
The final calculation result is given in Table2-2.
Table 2-2 Final Calculation Result
Basic
size
ES
EI
es
ei
Xmax(Ymax)
Xmin(Ymin)
TD
Td
Tf
Φ50
-0.036
-0.061
0
-0.016
-0.061
-0.020
0.025
0.016
0.041
Φ60
+0.030
0
+0.030
+0.011
0.019
-0.030
0.030
0.019
0.049
2.4 Series of Standard Tolerance
International tolerance grade specifies the tolerance with associated manufacturing
processes for a given dimension and can be calculated as
0.2 ITG 1
T  10

 0.45  3 D  0.001 D

(2-7)
where T is the tolerance in micrometers, D is the geometric mean dimension in
millimeters, and ITG is IT grade and is a positive integer.
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Most manufacturing processes have an IT grade designated for specification. IT
grades provide guidance for typical manufacturing process capability or how precise
one can except manufacture of a particular feature. When designing parts or
specifying mechanical tolerances, the size and location for a particular feature should
be determined. Using the designated international tolerance grade formula, the IT
grade required or typical to produce the part feature is assigned. For example, plastic
injection molding is determined to have an IT grade of 13 and a part needs an IT
grade of 5, engineering and design should consider an alternative and more capable
manufacturing process to produce the part and feature.
The permissible variation of size is called the tolerance. It is the difference between
the maximum and minimum permissible limits of the given size. It is the product of
the standard tolerance unit and grade of tolerance.
Standard Tolerance Unit: In a standard system of limits and fits, groups of
tolerances are considered as corresponding to the same level of accuracy for all basic
sizes. It is the name given to one standard series of tolerances calculated according to
certain law in terms of the basic size. Thus, it is a function of basic size and is
common to the two formulae defining the different grades of tolerance.
Grade of Tolerance: is an indication of the magnitude of tolerance. It may be divided
into 20 grades and each grade is represented by the Arabic numerals, from IT01, IT0,
and IT1…to IT18. Among them, IT01 possesses the highest precision and the least
tolerance, while IT18 possesses the lowest precision and the largest tolerance. The
lower the grade, larger will be the tolerance. Values of tolerance grade for some basic
sizes are given in Table2-3.
Table2-3 Basic Tolerances
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Example5: The basic sizes of holes are Φ16, Φ44, and Φ104, respectively. Their
tolerance value are 0.018, 0.025, 0.035when the tolerance grade of the hole is IT7,
respectively. Their tolerance value are 0.18, 0.25, 0.35when the tolerance grade of the
hole is IT12, respectively.
From the example, it can be seen that the larger the basic size, the lager is the
tolerance value, when the grade of tolerance is same. Besides, the lower the grade,
larger will be the tolerance value when the basic size is same.
2.5 Series of Basic Deviation
Basic Deviation: It is the upper or lower deviation used to determine the relative
position between the tolerance zone and zero line. Generally, the deviation nearer to
the zero line is treated as the basic deviation. It is the limit deviation that determines
the tolerance zone's location relative to the zero line in the GB/T 1800 series’ standard
limit and system of fits. There are 28 basic deviations in total for holes and shafts,
respectively. All of them are represented by Latin letters in which the capital letters
represent the holes, and the small letters represent the shafts. The distribution of basic
deviations for holes and shafts is shown in Fig.2-16 and the values of basic deviations
for shafts and holes are given in Table2-4 and Table2-5, respectively.
Figure 2-16 Series of Basic Deviation
Example6: Φ50H8, Φ50 is the basic size, H is the grade of basic deviation, and 8 is
the grade of standard tolerance. It is only used for holes.
Example2-7: Φ50f7, Φ50 is the basic size, f is the grade of basic deviation and 7 is
the standard tolerance. It is only used for shafts.
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Table 2-4 Basic Deviation Values of Shafts
19
Table 2- 5 Basic Deviation Values of Holes
2.6 Selecting the Limits and Fits
2.6.1 Selecting a Reference System
In order to obtain the different fits between the hole and shaft of the same basic size,
the fit system may be divided into two kinds, basic hole system and basic shaft
system.
Basic Hole System (shown in Fig.2-17): It is a system of fit in which the minimum
hole is taken as the basic size, and an allowance is assigned and tolerances are applied
on both sides of and away from this allowance. Holes in this system are called datum
holes. They are represented by the letter “H” and their lower deviations are always
zero.
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Figure2-17 Basic Hole System
Basic Shaft System (shown in Fig.2-18): It is a system of fits in which the maximum
shaft is taken as the basic size, and the allowance is assigned, and tolerances are
applied on both sides of, and away from this allowance. Shafts in this system are
called datum shafts. They are represented by the letter “h” and their upper deviations
are always zero.
Figure 2-18 Basic Shaft System
Principles of Selecting A Reference System：
The hole-based system is preferred. For small size holes of high-precision, applying
the hole-based system can reduce the numbers and specifications of specified value
cutting and measuring tools.
The shaft-based system is used in the following situations:
 The shaft is made of cold drawing bar stocks without machining;
 The shafts of the same size form different fits with different holes;
 The fit between the external ring for the rolling and the bearing housing adopts
shaft-based system, for rolling bearing is a standard part.
The related examples are given in Fig.2-19.
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Figure2-19 Examples of Selecting the Shaft-Based System
2.6.2 Selecting Grade of Tolerance
Basic principle for selecting the grade of tolerance is that choosing a low tolerance
grade as much as possible in the context of meeting the design requirements. In the
general machinery industry, the tolerance grade being applied is varied between IT5~
IT12. Tolerance grade of particular use can be determined in light of Tables 2-6 and
2-7.
Table2-6 Tolerance Grade for Long-Series Production Raw Castings
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Table2-7 Use Field of Individual Tolerances for the System ISO
IT01 to IT6
IT5 to IT12
IT11 to IT16
IT16 to IT18
IT11 to IT18
For production of gauges and measuring instruments
For fits in precision and general engineering
For production of semi-products
For structures
For specification of limit deviations of non-tolerated dimensions
2.6.3 Selecting Fit System
The guidelines for the selection of clearance fits are as follows:
 The fits H7-d8, H8-d9, and H11-dll are loose running fits, and are used for
plumber-block bearings und loose pulleys.
 The fits H6-e7, H7-e8 and H8-e8 are loose clearance fits and are used for
properly lubrication bearings, requiring appreciable clearances. The finer grades
are used for heavy-duty, high-speed bearings and large electric motors.
 The fits H6-J6, H7-f7 and H 8-f8 are normal running fits widely used for grease
or oil lubricated bearings, having low temperature rise. They are used for shafts
of gear boxes, small electric motors and pumps.
 The fits H6-g5, H7-g6 and H8-g7 are expensive from manufacturing
considerations. They are used in precision equipment, pistons, slide valves and
bearings of accurate link mechanisms.
The general guidelines for the selection of transition fits are as follows:
 The typical types of transition fits are H6-j5, H7-j6 and H8-j7. They are used in
applications where slight interference is permissible. Some of their applications
are spigot and recess of the rigid coupling and the composite gear blank, where
steel rim is fitted on ordinary steel hub.
 The fits H8/j7, H7/js6, H7/j6, and J7/h6 are small clearances or negligible
interference. The parts can be assembled or disassembled manually. They are
used for the easily dismountable fits of hubs of gears, pulleys and bushings,
retaining rings, and frequently removed bearing bushings.
 The fits H8/k7, H7/k6, K8/h7, and K7/h6 are small clearances or small
interferences. The parts can be assembled or disassembled without great force
using a rubber mallet. They are used for demountable fits of hubs of gears and
pulleys, manual wheels, clutches, and brake disks.
 The fits H8/p7, H8/m7, H8/n7, H7/m6, H7/n6, M8/h6, N8/h7, and N7/h6 are
negligible clearances or small interferences, and are mounting of fits using
pressing and light force. They are used for fixed plugs, driven bushings,
armatures of electric motors on shafts, gear rims, and flushed bolts.
The general guidelines for the selection of interference fits are as follows:
 The fit H7-p6 or H7-p7 results in interference, which is not excessive but
sufficient to give non-ferrous parts a light press fit. Such parts can he dismantled
easily as and when required, e.g., fitting a brass bush in the gearing tolerances.
23
 The fits H8/s7, H8/t7, H7/s6, H7/t6, S7/h6, and T7/h6 are the pressed fits with
medium interference. Assembling parts with such fits should use hot pressing or
cold pressing only with use of large forces. These fits are used for permanent
coupling of gears with shafts and bearing bushings.
 The fits H8/u8, H8/u7, H8/x8, H7/u6, U8/h7, and U7/h6 are the pressed fits with
big interferences. Assembling parts with such fits should use great forces under
different temperatures of the parts. These fits are used for permanent couplings of
gears with shafts, flanges.
The national standard indicated the general, common and preferred tolerance zones
for shafts and holes is shown in Table2-8.
Table2-8 Preferred Fits for Hole Basis System
Table2-9 Preferred Fits for Shaft Basis System
24
2.7 Marking Limits and Fits
(1) Marking in assembly drawings
There are two methods for marking the limits and fits in the assembly drawings,
which can be expressed as
a. Basic size
Hole basic deviation code
Shaft basic deviation code
IT
IT
,  40
H8
f7
b.
Basic size Tolerance zone code of the hole Tolerance zone code of the shaft 40 H 8 f 7
The related example s are given in Fig.2-20.
Figure2-20 Marking Examples
(2) Marking in the detail drawings
When marking the limits and fits in the detail drawings, two jobs should
be
performed, which are marking code and marking values. The related method is
described as follows:
1) Marking code:
The method for marking code can expressed as
Basic size Tolerance zone code
For example: 40H 8 ,  40 f 7
2) Marking values:
The method for marking values can expressed as
25
upper deviation
deviation
basic sizelower
0.039
For example:  400.025
0.050 ,  40 0
3) Comprehensive marking:
Thus, the comprehensive marking method can be described as
basic
size
Tolerance
zone code
upper deviation
lower deviation
0.039
For example:  40 f 7  0.025
,

0.050   40 H 8  0
Some marking examples are given in Fig.2-21.
Figure2-21 Marking Examples
Example8:
A fit D  d  80 , Xmax=0.012, and Ymax= -0.042. Determine the fit type, the basic
deviations of hole and shaft, and the grade of tolerance.
Solution:
Grade of tolerance: Tf = Xmax-Ymax =0.012-(-0.042) =0.054
Tf = TD + Td
Hole: IT7=0.030
shaft: IT6=0.019
hole basis system
basic deviation of hole: EI=0 ES=+0.030
basic deviation of shaft: Xmax=ES-ei ，ei=+0.018
basic deviation code of shaft is n
basic deviation of shaft: ei=+0.020, es=0.020+0.019=+0.039
the fit isΦ80H7/n6
26