Chapter4
Tolerance Rules
4.1 Terms
Actual Local Size (Da, da) : is the value of any individual distance at any cross
section of a size feature as shown in Fig. 4-1. The actual local size is a two-point
measurement, taken with an instrument like a caliper or micrometer, which is checked
at a point along the cross section of the part. A feature of size may have several
different values of actual local size.
Figure 4-1 Actual Local Size
Actual Mating Envelope (dfe): is defined according to the type of size feature being
considered. The actual mating envelope of shaft is similar perfect feature counterpart
of the smallest size that can be circumscribed about the feature so it just contacts the
surfaces at the highest points. For example, a similar perfect counterpart could be a
smallest cylinder of perfect form or two parallel planes of perfect form at a minimum
separation. It just contacts the highest points of the surfaces. Actual mating envelope
is a variable value and is derived from an actual part. See Fig.4-2 for examples. The
actual mating envelope of a shaft dfe can be calculated as
d f e  da  f
(4-1)
where da is the actual local size of a shaft and f is its form or position tolerance.
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Figure 4-2 Actual Mating Envelope of a Shaft
The actual mating envelope of a hole (Dfe) is a similar perfect feature counterpart of
the largest size that can be inscribed within the feature so that it just contacts the
surfaces at their highest points. A similar perfect feature counterpart could be a largest
cylinder of perfect form. It could also be two parallel planes of perfect form at the
maximum separation that just contact the highest points of the surfaces. Actual mating
envelope is a variable value and derived from an actual part. See Fig.4-3 for examples.
The actual mating envelope of a hole Dfe can be calculated as
D f e  Da  f
(4-2)
where Da is the actual local size of a hole and f is its form or position tolerance.
Figure 4-3 Actual Mating Envelope of a Hole
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Virtual Condition (VC): is the theoretical extreme boundary of a size feature
generated by the collective effects of MMC (the maximum material principle) and any
applicable geometric tolerances. It is a worst-case boundary generated by the
collective effects of a size feature at MMC or at LMC (the minimum material
principle) and the geometric tolerance for that material condition. The VC of a size
feature includes effects of the size, orientation, and location for the size feature. The
virtual condition boundary is related to the datums that are referenced in the position
tolerance used to determine the VC. Some examples of VC are shown in Fig.4-4.
Figure 4-4 Examples of Virtual Condition
Worst-Case Boundary (WCB): is a general term to refer to the extreme boundary of
a size feature that is the worst-case for assembly. Depending upon the part
dimensioning, a worst-case boundary can be a virtual condition, inner boundary, or
outer boundary.
Inner Boundary (IB): is a worst-case boundary generated by the smallest size feature
minus the stated form or position tolerance. Examples are given in Fig. 4-5.
Outer Boundary (OB): is a worst-case boundary generated by the largest feature size
plus the stated form or position tolerance. Examples are given in Fig. 4-5.
When a form or position tolerance that contains no modifiers in the tolerance portion
of the feature control frame is applied to a size feature, the inner or outer boundary of
size feature is affected. On an external size feature, the term "outer boundary" or
"worst-case boundary" is used. Outer boundary (OB) for an external size feature is the
worst case boundary generated by the largest feature (MMC) plus the form or position
tolerance. On an internal size feature, the term "inner boundary" is used. Inner
boundary (IB) for an internal size feature is the smallest feature (MMC) minus the
form or position tolerance.
In the case of an external size feature, such as a pin or a shaft, the OB is determined
by the following formula:
OB=MMC + Form or Position Tolerance, i.e., d MV  d M  t
(4-3)
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In the case of an internal size feature, such as a hole, the IB is determined by the
following formula:
IB=MMC - Form or Position Tolerance, i.e., DMV  DM  t
(4-4)
Figure4-5 Inner and Outer Boundary Examples
4.2 Tolerance Rules
In all cases, tolerances apply to features which are geometrical elements of a part,
such as: lines, planes, circles, cylinders, cones, spheres, helixes, tore, and
mathematically defined curves and surfaces (profiles). The main characteristics of a
feature and a geometry are its size, form, orientation, and location. Size tolerances are
referred to as dimensional tolerances, while form, profile, orientation, location and
run-out tolerances are referred to as form or position tolerances.
Several principles and good practices for specifying tolerances are either explicitly
defined or implied in the manufacturing tolerance standards. The principles help to
ensure that: the functional, manufacturing and inspection intents are clear, the
individual specifications form a consistent set of specifications with clear intent and
priority, the specifications are realistic in the sense that they are producible and
measurable, and unnecessary accumulation is minimized.
So, during the process of defining the tolerances of single parts or groups of
components, according to ISO 8015, it must always be decided if the independence
principle can allow for a toleranced size or if the envelope requirement has to be used.
Additionally it must also be decided if a deviation of form or orientation, location or
run-out has to be stated independently or if the maximum material requirement,
minimum material requirement or reciprocity requirement can be allowed for certain
form and position tolerances. Beside the previously discussed principles of
tolerancing, other conditions like the projected tolerance according to ISO 1101 or the
free state according to ISO 10579 are used in technical drawings (Fig.4-6). In this
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section, only principle of independency, envelope requirements, and maximum
material principle would be introduced.
Tolerance Rules
Principle of
Relevant
Independency
Requirements
Envelope
Maximum Material
Minimum Material
Reversible
Requirements
Principle
Principle
Requirements
Figure 4-6 Principle of Tolerance Rules (ISO 1101)
4.2.1 Principle of Independency
Each specified dimensional or geometrical requirement on a drawing shall be met
independently of other requirements, unless a particular relationship is specified. This
is the fundamental principle of tolerancing and it is named the “principle of
independency”. Therefore, where no relationship is specified, the form or position
tolerance applies regardless of feature size and the two requirements are treated as
being unrelated. Consequently, if a particular relationship of size and form (see the
envelope requirement), or size and location, or size and orientation is required, it shall
be specified on the drawing. When reference to the principle of independency is made
on the drawing, the dimensional tolerance, regardless of whether it is expressed as a
numerical tolerance or an ISO tolerance, does not control the form if the form
deviation is such that it will not be detected by two-point measurement.
Form deviations are controlled through form and position tolerances. Form and
position tolerances are used to control form, orientation and/or position on real or
abstract features, regardless of the feature size. The form and position tolerances will,
therefore, apply independently of the actual local sizes of individual features. The
form and position deviations may be at a maximum whether or not the cross sections
of the respective features are at the maximum material size. For instance, a shaft with
the maximum material size at any cross-section may have a lobed form deviation
within the circularity tolerance, and may also be bent by the amount of the
straightness tolerance, see Figs 4-7a and 4-7b.
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Figure 4-7 Principle of Independency: a) Drawing Indication; b) Interpretation
The linear and form and position tolerances are independent of each other, which
mean that the form and position tolerances apply regardless of the feature size. The
form and position deviations may be at maximum whether or not the cross-sections of
the respective features are at the maximum material limit/size (the size of the smallest
permissible hole or the size of the largest permissible shaft respectively). This is true
unless a particular relationship has been specified.
For the component using principle of independency, its worst-case boundary is the
boundary of limit sizes, i.e., the limits of sizes. Its qualified condition can be
described as
Hole : Dmin  Da  Dmax
shaft : d min  d a  d max
f t
f t
(4-5)
where f is the form or position error and t is the form or position tolerance.
Example1:
A part is shown in Fig.4-8. 1) Indicate the type of tolerance rule and boundary name;
2) calculate the boundary size and the value of the maximum straightness error; 3) if
the actual size Da =  20.01 and form tolerance f =  0.01, is the part qualified?
Solution:
Type of tolerance principle: principle of independency
Virtual condition: boundary of limit sizes, i.e., 20-20.033
Dmin  Da  Dmax
f t
20  Dmin  Da  20.01  Dmax  20.033 0.01  f  t  0.02
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The part is qualified.
Figure 4-8 A Part
4.2.2 Envelope Requirement
The envelope requirement is states that the real feature must not exceed the
geometrical ideal envelope with the maximum material limit within the mating zone.
The real size has to be as far away from the maximum material limit, as the expected
deviation of form will be. In practice, the application of the envelope requirement
means that if the actual size of the feature deviates from the maximum material limit,
the form deviations are allowed to be larger.
The envelope requirement may be applied to a single feature, for instance, a
cylindrical surface or a feature established by two parallel flat surfaces. The envelope
requirement is used for features which are to be mated. The envelope requirement
may be indicated either by the symbol
placed after the linear tolerance (Fig. 4-9a)
or in relevant cases by reference to an appropriate standard which invokes the
envelope requirement.
Fig.4-9a shows an example of the envelope requirement applied to a cylindrical
feature. The function requirements are:
 the surface of the cylindrical feature shall not extend beyond the envelope of
perfect form at maximum material size of Ø150;
 no actual local size shall be less than Ø149.96.
The drawing indication means that the actual part shall meet the following
requirements:
 each actual local diameter of the shaft shall remain within the size tolerance of
0,04 and may therefore vary between Ø150 and Ø149,96, see Fig.4-9b;
 the entire shaft shall remain within the boundary of the envelope cylinder of
perfect form of Ø150, see Fig.4-9c and 4-9d.
Hence it follows that the shaft shall be exactly cylindrical when all actual local
diameters are at the maximum material size of Ø150, see Fig.4-9e. On the drawings
where the principle of independency is not applied, it is applied the envelope
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requirement for all the dimensional and form and position tolerance (the envelope
requirement is not indicated). In this case, the tolerances are interpreted in the
following ways within the prescribed length:
 For holes: the diameter of the largest perfect imaginary cylinder which can be
inscribed within the hole so that is just contacts the high points of the surface
should not be smaller than the maximum material limit of size. The maximum
diameter at any position in the hole must not exceed the least material limit of
size.
 For shafts: the diameter of the smallest perfect imaginary cylinder which can be
circumscribed about the shaft so that it just contacts the high points of the surface
should not be larger than the maximum material limit of size. The minimum
diameter at any position on the shaft must not be less than the least material limit
of size.
Figure 4-9 Envelope Requirement: a) Drawing Indication; b, c, d, e) Interpretation
The above interpretation means that if the part is everywhere at its maximum material
limit, then the part should be perfectly round and straight, i.e. a perfect cylinder.
Unless otherwise specified and subject to the above requirements deviation from a
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perfect cylinder may reach the full value of the diametrical tolerance specified.
In special cases, the maximum form deviations permitted by the above interpretation
may be too large to allow satisfactory functioning of the assembled parts. In such
cases, separate tolerances shall be given for the form, e.g. separate tolerances on
circularity and/or straightness (see Fig.4-10). If we are adding on the previous
example a complementary form exigency-circularity tolerance (Fig.4-10a), which is a
compulsory restriction in comparison with the dimensional tolerance, the
interpretation in given in the Fig.4-10b.
Figure4-10 Envelope Requirement: a) Drawing Indication; b) Interpretation
For the component using the envelope requirement, its worst-case boundary is the
maximum material condition, i.e., the maximum material size DM, dM. Its qualified
condition can be described as
Hole : Dmin  DM  Dmax
shaft :
d min  d a  d max
D fe  DM
d f e  dM
(4-6)
where Dfe and dfe are the actual mating envelope for hole and shaft, respectively.
Limitation of Envelope Requirements
A part often contains multiple features of size. Envelope rule does not affect the
location, orientation, or relationship between features of size. Features of size shown
perpendicular, symmetrical, or coaxial must be controlled for location or orientation
to avoid incomplete drawing specifications. Often, implied 90 angles are covered by a
general angular tolerance note or by an angular tolerance in the drawing title block.
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Example2:
A shaft is shown in Fig.4-10. 1) Indicate the type of tolerance rule and boundary name;
2) calculate the boundary size and the value of the maximum straightness error; 3) if
the actual size da= Φ19.98, form tolerance f =0.015, is the part qualified?
1) envelop requirement, maximum material boundary
2) dM=20mm,
f=|dM -dL|=20-19.979 =0.021
3)
d fe  d M d max  d a  d min
d fe  d a  f  19.995  d M  20
d max  20  d a  19.98  d min  19.979
Part is qualified.
Figure4-10 A Shaft
4.2.3 Maximum Material Principle
The limit of size, together with geometrical form or position of a feature, is factors of
the maximum material principle, and its application is restricted to those features
whose size is specified by toleranced dimensions incorporating an axis or median
plane. It can never be applied to a plane, surface, or line on a surface. The
characteristics to which it can be applied are as follows: straightness, parallelism,
squareness, angularity, position, concentricity, and symmetry. The characteristics to
which the maximum material condition concept cannot be applied are flatness,
roundness, cylindricity, profile of a line, profile of a surface, and run-out.
Fig.4-11(a) shows a typical drawing instruction where limits of size are applied to a
pin, and in addition a straightness tolerance of 0.2 is applicable at the maximum
material condition. Fig.4-11(b) shows the condition where the pin is finished at the
maximum material condition with the maximum straightness error. The effective
assembly diameter will be equal to the sum of the upper limit of size and the
straightness tolerance. The straightness error is contained within a cylindrical
tolerance zone of 0.2. To provide the same assembly diameter of 20.7 as shown in
Fig.4-11(c) when the pin is finished at its low limit of size of 20.0, it follows that a
straightness error of 0.7 could be acceptable. This increase may in some cases have no
serious effect on the function of the component, and can be permitted.
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(a)
(b)
(c)
Figure4-11 Maximum Material Condition Applied to Straightness
For the component using the maximum material principle, its worst-case boundary is
the inner boundary or the outer boundary, i.e., DMV, dMV. Its qualified condition can
be described as
Hole : Dmin  DM  Dmax
shaft :
d min  d a  d max
D fe  DMV
(4-7)
d f e  d MV
where Dfe and dfe are the actual mating envelope for hole and shaft, respectively. DMV
and dMV are the size of inner boundary and the size of outer boundary, respectively.
+0.021
φ40 0
D fe  Da  f  40.015  0.015  40
Component is qualified.
φ30 0
Dmin  40  Da  40.015  Dmax  40.025
+0.025
Example3:
A component is shown in Fig.4-12. 1) Indicate the type of tolerance rule and
boundary name; 2) calculate the boundary size and the value of the maximum
straightness error; 3) if the actual size Da= Φ40.015, form tolerance f =0.015, is the
part qualified?
1)the maximum material principle, IB boundary
2) DMV  DM  t
= 40-0.02=39.98
φ0.02 M A
f = t+|DM-DL| = 0.02+0.025 =0.045
3) D fe  DMV Dmax  Da  Dmin
A
Figure4-12 A part
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