Tolerances

Calculating Locational Tolerances

Howard Gibson CET

2000 August 27

Contents

1 Introduction 1

1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Copyright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Change History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Modelling 1

2.1 Bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Screws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Positional Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Fastener Configurations 3

3.1 Only Two Fasteners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Four Fastener Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Datum Dimensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Pitch Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 A more elaborate pitch circle model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Positional Tolerances 10

4.1 Calculation of Positional Tolerance of Datum Dimension . . . . . . . . . . . . . . . . 104.2 A More Precise Geometric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Example 12

6 Closing Remarks 12

i

List of Figures

1 Typical Bolted Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Typical Screwed Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Attachment with two bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Attachment with Four Bolts in a Rectangle . . . . . . . . . . . . . . . . . . . . . . . 5

5 Bolts located from a Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 Bolts arrayed on a Pitch Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

7 Bolt Located by a Positional Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8 Geometric Dimension on Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

ii

11 Introduction

1.1 Objective

Location tolerances are required to ensure that parts, and their fasteners, fit together and thatfeatures are located to within specification. What follows is a systematic procedure for determininglocational tolerances for bolts, screws, pins, shafts, and features embedded on the parts. Thisdocument is intended as a guide for designers, drafters and drawing checkers.

1.2 Copyright

This document is copyright c 2000 August 27 by Howard Gibson. You may copy this documentonto bulletin boards, web pages and other computer media, as long as the file is unaltered, and thedistribution is not-for-profit. All other rights are reserved.

1.3 Change History

1998 Sep 14: First issue.

1998 Dec 01: Clarified some text. Expanded Positional Tolerances section a bit.

1999 Aug 24: Added a web page reference. Typo correction.

2000 Jul 20: Cleaned up the HTML formatting a bit. No change in text.

2000 Aug 27: Clean up HTML some more, finally getting background colour to work. Maximumerror for datum dimensioning was 45 opposite to each other. The section of precise geometrictolerancing make a little clearer.

2 Modelling

When parts are held together by more than one fastener, the tolerances for the holes must ensurenot only that the parts are located to within specification, but also that all the fasteners can beinstalled. Analysis will consist of determining the maximum possible location error that will allowthis, and then working out the equivalent drawing tolerance. It will be assumed that all holesare drilled, punched, tapped or whatever, prior to assembly. If the tolerances generated by thefollowing calculations are not reasonable, drilling at assembly may be the only solution.

The analysis will cover a series of fastener configurations and dimensioning procedures. For thepurposes of building analysis models, fastener assemblies can be split up into two categories, whichwill be termed here bolts and screws. For each of these, we can determine a maximum ac-ceptable error, and for each design configuration, and for each style of tolerance application on thedrawing, we can work out the tolerance.

2 2 MODELLING

C2

DBOLT

DHOLE

C2

DHOLE

C

Nominal Position

Figure 1: Typical Bolted Connection

2.1 Bolts

Fasteners pass through clearance holes in both parts. The obvious example of this is a nut and boltassembly, but this could also be a cotter pin assembly or a pre-drilled rivet assembly. The boltscould be clamping more than two parts at a time.

See Figure 1. It will be assumed for analysis purposes, that bolts are located precisely at theirnominal position. The two clearance holes are shifted off the nominal position such that their edgesjust touch the bolt, on opposite sides. This is the maximum acceptable error condition. If theclearance in each hole is c, then. . .

Maximum allowable shift of holes for bolting:

b =c

2(1)

2.2 Screws

Fasteners pass through a clearance hole in one part, and are solidly located in the other part. Thesecould be screws sitting in tapped holes, or they could be dowel pins, or features integral to one ofthe parts. As with a bolted assembly, a screw could be holding more than one piece to its base.

See Figure 2. It will be assumed that the screw and clearance hole have moved in opposite directions,such that the screw is touching the edge of the hole. If c is the clearance in each hole, then. . .

Maximum allowable shift of holes for screwing:

s =c

4(2)

2.3 Positional Error 3

DHOLE

C4

DSCREW C

Nominal Position

Figure 2: Typical Screwed Connection

A screw assembly allows only half the error of a bolted assembly. This is something to consider ifthe location tolerances are difficult to maintain.

2.3 Positional Error

For two dimensional configurations, it will usually be assumed that the maximum error occurs at a45 angle. The assumption is that the fabricators capability is equal in both the x and y directions.If this assumption is not true for a given application, then the angle, and the resultant calculationsand tolerances can be fudged a bit.

3 Fastener Configurations

The tolerance that must be applied on a drawing feature depends on the configuration of thefasteners and on the dimensioning procedure. Most articles on ASME Y14.5M-1994 GeometricDimensioning and Tolerancing point out that GD&T provides a maximum of allowance for a givenpart. This is important if the tolerance to be achieved is difficult, and if an increase in allowance canreduce costs. If tolerances are fairly easy, then the cost of preparing and reading the drawings mustbe considered. Regardless of how the tolerances are applied, GD&T provides a precise interpretationof how the tolerances are to be determined from the dimensions.

3.1 Only Two Fasteners

Parts are assembled with two fasteners. It is assumed that the parts are not located accurately toeach other. On the drawings, the holes are dimensioned from each other, rather from a datum.

4 3 FASTENER CONFIGURATIONS

C2

DIM - tDIM + t

Figure 3: Attachment with two bolts

In Figure 3, the maximum error occurs at both holes, so that tolerance is double the error. . .

tb = 2b = 2 c2

Tolerances for dimensions between two bolts

tb = c (3)

ts = 2s = 2 c4

Tolerances for dimensions between two screws:

ts =c

2(4)

3.2 Four Fastener Pattern

Parts are assembled with four fasteners. Again, the relative location of the parts is not critical. Onthe drawing, the holes are dimensioned with respect to each other as shown in Figure 4.

The holes in Figure 4 are shown shifted the maximum distance, at a 45 angle. Again, the erroroccurs at each hole.

tb = 2 bsin 45 = 2c/2

2

3.2 Four Fastener Pattern 5

C2

VERT + t VERT - t

HORIZ - tHORIZ + t

Figure 4: Attachment with Four Bolts in a Rectangle


C2

VERT - tHORIZ - t

HORIZ + tVERT + t

Figure 5: Bolts located from a Datum

The tolerance for dimensions between four bolts:

tb =c

1.414(5)

ts = 2 ssin 45 = 2c/4

2

The tolerance for dimensions between four screws:

ts =c

2.828(6)

3.3 Datum Dimensioning

Parts are assembled with any number of fasteners. On the fabrication drawings, all the holes aredimensioned from arbitrarily selected X and Y datums 1. For one-off parts, this is popular withmachine shops (and therefore cheaper) because only one set-up is required to locate all the holes.If the two parts are to be located accurately the holes must be registered to some feature otherthan another fastener hole. On any non-rectangular hole pattern with more than two holes, thedimensioning is effectively datum, even if everything is referenced from one of the holes.

1According to the Oxford Concise Dictionary, the plural of datum is data. This doesnt parse very well.Fortunately, Websters Ninth New Collegiate Dictionary allows datums.

3.4 Pitch Circles 7

The model assumes that two holes are located from datums Each hole can shift in two dimensions,even if the holes are arrayed in a single line. It is assumed that the maximum error occurs whenboth holes shift to the maximum allowed, in opposite directions.

2 t2b = b2 =(c

2

)2

t2b =c2

8

The tolerance for a datum dimension to a bolt hole:

tb =c

2.828(7)

2 t2s = s2 =(c

4

)2

t2s =c2

32

The tolerance for a datum dimension to a screw hole:

ts =c

5.657(8)

3.4 Pitch Circles

The fasteners are located at specified positions around a circle, as shown in Figure 6. This model issimilar to the datum dimensioning in that each fastener is located from a datum by two dimensions.

For a bolted assembly, the total error must not exceed b. This error is a combination of t, ,and D. Since the dimension tolerance t acts on the diameter, the effective allowance due to thespecified t is t/2. . .

b2 =(c

2

)2=(tb2

)2+(D

2tan b

)2The simplest solution to this is to assume that the maximum allowable error acts at a 45 anglefrom the radial. In this case, the effective diameter error equals the effective angular error. . .

12

(c

2

)2=(tb2

)2=(D

2tan b

)2


C2

D + t

D - t

+

Figure 6: Bolts arrayed on a Pitch Circle

c2

8=t2b4

=14D2 tan2 b

c2

2= t2b = D

2 tan2 b

t2b =c2

2and. . . tan2 + b =

c2

2D2

tb =

c2

2and. . . + b = arctan

c2

2D2

For a bolted assembly arranged on a pitch circle, the diametral and angular tolerances are. . .

tb =c

1.414and. . . b = arctan

(c

1.414D

)(9)

3.5 A more elaborate pitch circle model 9

For a screwed assembly, the total error must not exceed s. . .

s2 =(c

4

)2=(ts2

)2+(D

2tan s

)2Again, we assume that the maximum allowable error occurs at 45.

12

(c

4

)2=(ts2

)2=(D

2tan s

)2

ts =

c2

8and. . . s = arctan

c2

8D2

For a screwed assembly arranged on a pitch circle, the diametral and angular tolerances are. . .

ts =c

2.828and. . . s = arctan

(c

2.828D

)(10)

3.5 A more elaborate pitch circle model

In the above system, the angular tolerance must be tightened as the diameter increases. At somemagnitude of diameter, this will be too tight for fabrication at an acceptable cost. There are severalways around this.

Dont specify pitch circles. Use datum dimensioning. Linear dimensions are not sensitive toincreasing size. Fabricators often re-calculate pitch circles as datum dimensions anyway, sincethis is more suitable for their machinery. Datum dimensions are also easier to inspect.

Replace the holes with slots, with the long dimension arranged along the pitch circle. This iseconomical if the part is being punched.

Fudge the above calculations by opening up the angular tolerance a bit and closing thediametral tolerance to make up for it.

If on a bolted pitch circleD

2tan b < b =

c

2. . .

. . . then you can select a slightly larger angular tolerance and recalculate the diametral tolerance. . .

. . .

(c

2

)2=(tb2

)2+(D

2tan b

)2

10 4 POSITIONAL TOLERANCES

(tb2

)2=(c

2

)2(D

2tan b

)2

tb = 2

(c

2

)2(D

2tan b

)2If on a screwed pitch circle

D

2tan s < s =

c

4. . .

. . . then you can select a slightly larger angular tolerance and recalculate the diametral tolerance. . .

. . . ts = 2

(c

4

)2(D

2tan s

)2You dont gain much by all this calculation. If you are having problems, you should really considerdatum dimensions.

4 Positional Tolerances

4.1 Calculation of Positional Tolerance of Datum Dimension

A geometric tolerance specifies a diameter centred upon the nominal dimension, within which thecentre of the hole or feature must be located. The positional tolerance comes from a datum of somesort, even if this is only the base surface parallel to the surface the hole is on.

See Figure 7. Let G be the geometric tolerance for a bolted assembly.

Gb = 2b = 2(c

2

)For a bolted assembly, the geometric tolerance is . . .

Gb = c (11)

For a screwed assembly. . .

. . . Gs = 2s = 2(c

4

)For a screwed assembly, the geometric tolerance is. . .

Gs =c

2(12)

4.2 A More Precise Geometric Model 11

C2

G

Figure 7: Bolt Located by a Positional Tolerance

A B C0 MMMCLMC

Figure 8: Geometric Dimension on Drawing

4.2 A More Precise Geometric Model

Take a look at Figure 8. The positional tolerance of 0 looks horrifying, especially if it is done ona casting or weldment where accurate fabrication is difficult. Actually, it is a precise descriptionof the required geometry. Take the case that you are providing clearance for an M6 bolt. Basedon the bolt model described earlier, we locate the bolt at precisely the nominal position. For astandard bolt, the maximum allowable diameter is the nominal diameter. Therefore, a 6mm holelocated precisely at nominal provides the required clearance. If the hole is larger than 6mm, thenit can shift up half the clearance, as calculated earlier. The positional tolerance of 0 in Figure 8applies only at the maximum material condition, as indicated by the circle M..

The model for screws is more complicated, since we have two different kinds of hole. The maximumand least material conditions have no relevance for a tapped hole or for a hole for a press-fit dowelpin. We have to allow for some movement by applying a geometric tolerance G.

For the clearance hole, we can apply the same geometric tolerance as shown in Figure 8, but we willhave to increase the minimum dimension (maximum material condition) to account for the screwdiameter, plus the locational tolerance.

For any clearance hole for a screw...

MMC = DIA+G and LMC > MMC, obviously

12 6 CLOSING REMARKS

If our screw is actually a feature integral to a fabricated part, then the analysis for bolts is validfor both parts. The female features MMC is the maximum diameter, obviously. You can use theexact dimension format for both pieces.

5 Example

Take the case that we have a 1/4-20UNC screw, and a clearance hole of 9/32 DIA. Work out thetolerance for a datum dimension, and for a GD&T positional tolerance.

c = 9/32 1/4 = .03125in.

Datum Tolerance: ts = c/5.657 = .03125/5.657 = .0055 .005inPositional Tolerance: Gs = c/2 = .03125/2 = .0156 .015inFrom this, we can see that the datum tolerance of .005 allows the same maximum error as apositional tolerance of .015. The positional tolerance zone works out to .0152pi/4 = 1.8104in2.The tolerance zone of the datum tolerance is (2 .005)2 = 1 104in2.For a given acceptable error, the GD&T positional tolerance provides almost double the allowancefor locating holes. If your manufacturing process is marginally capable of meeting tolerances, thiswill affect your scrap rate. Positional tolerances are also nice on drawings for formatting reasons,since you are attaching the location tolerance to the hole specification, rather than to a dimensionline, which may point to several different holes, with different tolerance requirements.

The old, linear tolerances are still simpler on the drawing, and the extra allowance is irrelevant ifyour manufacturing process easily exceeds the required accuracy.

6 Closing Remarks

One thing to consider when applying tolerances to fabrication drawings, is the actual capabilityof your fabricators. The author has inspected parts fabricated in a machine shop. The observedlocation errors for holes were typically within 0.003 in. (

.006). This was close to the accuracy of

the vernier equipment that was used, so a significant part of this was measurement error.

The author has not checked sheet metal, castings or weldments with similar thoroughness.

He has been told by a sheet metal fabricator, that sheet metal holes can be located to within .015with respect to a folded edge. This fabricator is skilled, so other shops may not be as good.

Check out the D.S.M. Manufacturing Companys web page for more sheet metal tolerances. Fortheir sheet metal, they quote .003in for hole diameters, and .005in for hole to hole locations in

13

parts that do not have too many holes punched in them. They would prefer it if designers allowed.010in. Hole to fold tolerances should be .015in, and fold to fold tolerances should be about.020in.The D.S.M.s web page is at. . .

http://www.precisionsheetmetal.com/data.htm

One objective of toleranceing, particularly geometric toleranceing, is to control costs by allowing asmuch variance as possible. This can reduce scrap rates by increasing the probability that fabricatedparts conform to specification. This might reduce fabrication costs by allowing the fabricator towork faster, or use a cheaper manufacturing process.

Another possibility is that the standard manufacturing process can easily achieve the specifiedtolerances. In this case, a complicated drawing may increase the cost as the fabricator takes theextra time required to interpret the dimensions, and for that matter, as the designer takes theextra time required to produce the drawing! This is particularly true for one-off items, and foritems requiring complete inspection, and any other situation that imposes frequent reference to thefabrication drawings.

The designer must balance the cost of managing the drawings against the need for loose, easily metspecifications.

IntroductionObjectiveCopyrightChange History

ModellingBoltsScrewsPositional Error

Fastener ConfigurationsOnly Two FastenersFour Fastener PatternDatum DimensioningPitch CirclesA more elaborate pitch circle model

Positional TolerancesCalculation of Positional Tolerance of Datum DimensionA More Precise Geometric Model

ExampleClosing Remarks

Documents

Tolerances