Geometric Dimensioning and Tolerancing: Why ?

Geometric Dimensioning and Tolerancing (GDT) is a method for precisely defining the geometry of mechanical parts. It introduces tools which allow mechanical designers, fabricators, and inspectors to effectively communicate complex geometrical descriptions which are not otherwise able to be described in a defined language.  

Figure 1.1 A good example of why GDT is needed is the automobile stamped chassis shown in Figure 1.1. The rear quarter panel must fit snugly in order to allow spot welds and in this competitive business, cosmetic appearance and noise abatement are critical. Without GDT, geometric description of this assembly is difficult if not impossible. Computer modelling of these complex surfaces continues to increase the complexity of interface shapes.

Illustrated in Figure 1.2 is an imprecise sand-cast sewer termination fitting. Its hole pattern must mate with the corresponding pattern of the equally imprecise mating pipe. Flatness is also an issue with these rough-surfaced parts. Inspectors must be able to pass as many parts as possible without sacrificing fit. Tens of thousands of the parts are produced. GDT allows fabricators, inspectors, and assemblers to match covers with equally imprecise pipes.

Figure 1.2

Geometric Dimensioning and Tolerancing is a vast language of which there are many facets. However, what is commonly used is a small subset of the total. This subset is based on concepts which MUST be learned in order to progress further. Without a solid understanding of these fundamentals, one cannot gain a firm grasp of later topics. We will present the most essential (and often misinterpreted) topics in a step-by-step fashion, starting with a simple two-dimensional case. After the 2D case has been understood, the full three-dimensional geometry will be described. We also include common areas of confusion and a reference section, but at this point the primary objective is to explain the fundamentals. Please select "2D DATUMS" from the menu bar to the left to continue.  History of Geometric Dimensioning and Tolerancing  

Geometric Dimensioning and Tolerancing symbols have been in use since at least the turn of the century. GDT was especially important during the Second World War in relation to extremely high volume production of Liberty Ships, aircraft, and ground vehicles. The automotive industry, with its high volumes, has also benefited from GDT. The computer industry, in particular mass storage manufacturers, have used GDT extensively to increase their yields of high-volume and low-margin hard disk drives. However, as with most engineering and scientific methodologies, GDT was not rigorously established and documented until later in the twentieth century. The American National Standards Institute publication in 1982 of ANSI Y14.5M-1982 was a turning point in the rigorous, unambiguous standardization of the methodology. 2D Datum Examples

It has been found that the most powerful (yet often misunderstood) use for GDT is in the area of sizing and positioning of holes which must mate with shafts. We will start with a two-dimensional geometry and progress to three dimensions.  Figure 2.1 Pictured in Figure 2.1 is a two-dimensional part. The roughness of its edges has been greatly exaggerated

in order to clarify the discussion. It might be thought of as a part with plane dimensions large compared with its dimension into the page. For example, this might be a sheet metal part which is .125 inches thick with a length and width of about ten inches.In this example we will determine the position of the hole and in the process, find out why mutually-perpendicular ordered datums must be defined in order to define the position. Figure 2.1 shows the part with possible ways to define the location of the hole. As has been made apparent, the linear dimensions originate from the sides of the part, but it is not clear from where on the sides the dimensions should begin. (We consider where the center of the hole is later.) The method of fixturing the part for measurement of its features is critical since many different groups of people will be measuring the part: design, fabrication, fabrication inspection, purchasing inspection,

etc. Without a common agreement as to how the part will be measured, measurements become meaningless.

Figure 2.2 One way to define where the dimensions should originate is shown in Figure 2.2. A steel straight edge can be used to define a line for the two edges. One problem with this method is that the two defined lines are not necessarily perpendicular, as

shown in the figure. Without perpendicularity, the part dimensions do not agree with print dimensions, since print dimensions are assumed to be perpendicular.  We can force the two defined lines to be perpendicular to each other by using a right-angle straight edge, as shown in Figure 2.3. Now when the part is pushed against these two edges so that it cannot move (rock), the two edges can be used as mutually perpendicular datums. However, as Figure 2.4 shows, the part position with respect to these two datums is ambiguous. The final orientation of the part depends upon which side contacts first.

Figures 2.3 and 2.4

Figure 2.5 illustrates an ordered datum scheme that prevents confusion. The first side that is pressed against one of the edges will contact at the two highest points of the part edge. The part now only has one degree of freedom left: it can slide back and forth against the straight edge.

Figure 2.5 Once we butt the perpendicular side of the part against the corresponding straight edge, we will have completely constrained the position and orientation of the part in 2D space, as shown in Figure 2.5. This second side contacts its straight edge at one high point since it is not able to rotate to contact more than one high point. Figure 2.5 labels the datums as A and B, in the order of fixturing hierarchy. These datums are referred to as "functional datums" since they contact the part and are physical hardware used, for example, on a factory floor.

With this ordering of datums, we have clearly positioned the part in 2D space. As we have seen, this clear fixturing of the part is critical since it defines from where features are to be dimensioned on the part. Please continue by selecting "2D Hole/Shaft" from the menu on the left.

SIDE NOTE: Since this is a two-dimensional example, the part has three degrees of freedom: two translational and one rotational. The reason for the first two high points of contact has to do with the rotational degree of freedom of the part: one point allows the part to rotate, two does not. The single point of contact for -B- is due to the fact that once the part can only slide along -A- without rotating, a single high point will contact first on -B-. The part cannot rock, so a second contact point does not occur. 2D Hole Positioning

Figure 2.6a: Traditional Plus/Minus Tolerancing Now that we have rigorously defined how we fixture the part for dimensioning features, we can define where the hole is. Figure 2.6a shows two dimensions which show where the hole is with respect to datums A and B. Plus/minus tolerances are also shown, as is a square tolerance zone within which the center of the circle must lie. Although it may seem that we are not using GDT here, without the GDT datums, the two dimensions shown are ambiguous.

However, the plus/minus tolerance zone is not clearly defined since we do not know what is meant by the "center of the circle". In addition, a square tolerance zone is typically not what is needed for defining the position of a hole with respect to a mating shaft, as shown in Figure 2.7a.

Figure 2.7a Figure 2.6b

Figure 2.6b illustrates an isolated case where a square tolerance zone would be appropriate.  Figure 2.7 : GDT of the Same Hole Figure 2.7 illustrates the equivalent GDT tolerancing of the hole position. The tolerance zone within which the center of the circle must lie is circular rather than rectangular. We will later learn how the size and position of this circular zone is defined based upon the symbols shown. In GDT, the "center of the circle"

is defined as the center of the best-fit circle determined by the actual hole. Coordinate Measuring Machine (CMM) equipment typically uses at least three points on an actual hole to define the best-fit circle, as illustrated in Figure 2.7b. Figure 2.7c shows the CMM probe touching the side of the hole for a data point. The CMM machine of course uses the A and B functional datum planes as its position and orientation references. A big advantage of GDT is that the circular tolerance zone contains 57% more area than an equivalent square tolerance zone. The largest deviation from true position occurs on the diagonals of a square, and the circle meets this, while providing 40% more possible deviation along the vertical and horizontal. Therefore, more parts can be accepted by inspection. With the square tolerance zone, parts that can fit are rejected since typically only the vertical and horizontal location deviations are checked.  

Figure 2.7b : Best Fit Circle Center

Figure 2.7cThe 7.5 and 3.0 dimensions in Figure 2.7 do not have

attached tolerances for a reason. They are called basic dimensions and represent the exact position of the center of the circular tolerance zone within which the center of the circle must lie. They can be recognized as basic dimensions because they are box framed. The diameter of the circular tolerance zone comes from the feature control frame which is below the 2.5 hole diameter dimension. The diameter of the tolerance zone in the feature control frame is 0.5 inches. The first symbol in the frame designates the tolerance as a positional tolerance. The -A- and -B- are the GDT datums to which the zone refers. For a two dimensional problem, the importance of -A- and -B- is not apparent, but we will see in the 3D example how they come into play. For now, let us notice that the tolerance zone location is located via -A- and -B-. The ±0.2 tolerance on the 2.5 hole diameter allows the diameter of the hole to vary from 2.3 to 2.7, but the center of the hole must still lie within the circular tolerance zone described above. The ± 0.2 tolerance will be discussed further under bonus tolerancing.  Figure 2.8 There is another way in which the circular tolerance zone for the hole can be interpreted. Figure 2.8 shows this tolerance zone as being the annular "racetrack" formed by two circles centered at 7.5, 3.0, nominally 2.5 in diameter, and ±.125 above and below 2.5 in diameter. If the actual hole profile is within this "racetrack", it is very similar to its center being within the 0.5 diameter circular tolerance zone. This interpretation of hole positional tolerances is common and probably originated before CMM machines allowed finding the center of the best fit circle. This interpretation lends itself to inspection with calipers, gauge pins, and the like. However, in cases where extreme precision and the utmost certainty of geometry are required, the true GDT (ANSI Y14.5) definition should be used.  Now that we have observed how GDT can clearly define hole position, let us move on to bonus tolerancing. If you would like to skip bonus tolerancing for now and move on to the 3D case of what we have discussed, see 3D Datums. 2D Shaft Bonus Tolerance

GDT dimensioning of holes (and shafts) provides a powerful method for increasing inspection yield without trial and error fitting or binning. This method is what is known as bonus tolerancing. Figure 2.9 illustrates the possible size variation of the hole from the 2D Hole section. It does not, however, show the variation in position of the hole that is shown in Figure 2.10, for a nominal hole size. Figure 2.9(left) and Figure 2.10

Combining the two types of variation, we get Figure 2.11, which shows the quite large envelope which results. If the hole becomes larger, why not widen the tolerance on the hole position since the mating part now has a larger target ? This is what bonus tolerancing provides.  

Figure 2.11Figure 2.12 is the same hole that we have been discussing, but with an added symbol in the feature control frame. The M with a circle around it stands for Maximum Material Condition (MMC). For a hole, this is the smallest possible size, and for a shaft, it is the largest. In other words, for mating holes and shafts, MMC is the tolerance condition where fit is most difficult.

If the hole increases in size from MMC, leeway in the position of the hole becomes available. This addition to the positional tolerance zone diameter is defined as the amount that the hole diameter increases over MMC.

Figure 2.12If no MMC symbol is present, the tolerance is assumed to apply at Regardless of Feature Size (RFS) and there is no bonus tolerance. The Least Material Condition (LMC) also exists, symbolized by a circled L, but we have not found it to be used very often.

For a continuation of this discussion concerning virtual condition, please see 2D Bon Tol Example.

2D Hole/Shaft Bonus Tolerance Example

GDT dimensioning of holes (and shafts) provides a powerful method for increasing inspection yield without trial and error fitting or binning. This method is what is known as bonus tolerancing.  In the following series of figures, the captions suffice to explain the logic. What is being shown is how increasing the size of a hole above MMC allows a bonus in the size of the positional tolerance zone.

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

Figure 2.17

For a continuation of this discussion concerning virtual condition, see 2D Virtual Condition. 2D Virtual Condition

The Virtual Condition of a feature is a concept used to describe the worst-case envelope which either of two features must lie within in order to mate acceptably. For a shaft that fits into a hole, the shaft virtual condition must be smaller than the hole virtual condition.  For an external feature of size, such as a shaft, the virtual condition is equal to the size at MMC plus the size of the tolerance zone. For the shaft in Figure VC1.1, the diameter of the virtual condition is the diameter of the MMC shaft plus the diameter of the position tolerance zone.

Figure VC1.1

For an internal feature of size, such as a hole, the virtual condition is equal to the size at MMC minus the size of the tolerance zone. For the hole in Figure VC1.2, the diameter of the virtual condition is the diameter of the MMC hole minus the diameter of the position tolerance zone.

Figure VC1.2

Figure VC1.3 shows the shaft and hole virtual conditions superimposed. Since the shaft virtual condition is smaller than the hole virtual condition, the two parts will always mate.

Figure VC1.3

 In summary, the way to calculate virtual condition (VC) for a shaft and hole is: SHAFT VC = MMC diameter + Position Tolerance Zone DiameterHOLE VC = MMC diameter - Position Tolerance Zone Diameter  Virtual condition is extremely useful in the design of functional gauges. A functional gauge made to virtual condition will ensure that a part will always mate with its counterpart. For a description of how a real-world functional gauge works, please see 3D Functional Gauge. To start from scratch in 3D, go to 3D Datums. 3D Datums

We will now look at the three-dimensional (3D) case. Figure 3.1 illustrates a three-plane fixture for determining the position and orientation of a part for measurement. It is analogous to the L-bracket used for the two-dimensional(2D) case.  Figure 3.1The 3D part is also shown, its roughness exaggerated for clarity. The -A- datum will be contacted by the three high points of its corresponding surface, as illustrated. Three points determine a plane, so that the highest three points of the surface will position the part so that it can now only slide along the -A- datum. From here, the degrees of freedom of the part are similar to those for the 2D part: one rotational and two translational. As with the 2D part, the next datum will contact the two high points of the B surface. Now the part can only slide along -A- and -B- in a direction perpendicular to -C-. As with the 2D case, one high point of contact with

-C- completes the fixturing of the part in space with respect to the datums. Measurements can now be made from the datums with full confidence that whenever or wherever the part is measured, the numbers will be the same. 3D Hole

We will now demonstrate the dimensioning of a hole in GDT. As was seen in the 3D Datums section, the part that we are looking at is very much like the one in the 2D section, only we have added more thickness into the page to make the part three-dimensional. The plus/minus dimensioning in Figure 3.2 and GDT dimensioning in Figure 3.3 of the hole in this 3D part look exactly as they did in the 2D case, only the interpretation is slightly more extensive.  

Figure 3.2

Figure 3.3

The tolerance zone has now gained a dimension to become a 3D cylinder, as shown in Figure 3.4. The basic dimensions locate the central axis of the tolerance zone, as with the central point in 2D. In addition, the tolerance zone is "true" with respect to A, B, and C: The central axis is parallel to A and B and perpendicular to C. As shown in Figure 3.5, the central axis of the actual hole must lie completely within the tolerance zone cylinder.  Figure 3.4

 

 

 

             Figure 3.5

The central axis of the cylinder is defined as follows: At any cross-section of the hole perpendicular to the datum axis, the center of the cross-section circle is defined as it was in the 2D case. This is illustrated in Figure 3.6.

Figure 3.6

Bonus tolerances for the 3D case are exactly like those of the 2D case, but now the tolerance zone is a cylinder rather than a circle.See 3D Bonus Tolerance for details. 3D Hole Bonus Tolerance

As we saw in the 2D section, GDT dimensioning of holes (and shafts) provides a powerful method for increasing inspection yield without trial and error fitting or binning. This method is what is known as bonus tolerancing. The following figures are from the 2D section, but now have a third dimension out of the page. Since the tolerance zones and MMC envelopes form perfect right circular cylinders, we can use projections of the 2D figures. Figure 2.9 illustrates the possible size variation of the hole from the 2D Hole section. It does not, however, show the variation in position of the hole that is shown in Figure 2.10, for a nominal hole size.

Figure 2.9(left)Figure 2.10(right)

Combining the two types of variation, we get Figure 2.11, which shows the quite large envelope which results. If the hole becomes larger, why not widen the tolerance on the hole position since the mating part now has a larger target? This is what bonus tolerancing provides.

Figure 2.11

Figure 2.12 is the same hole that we have been discussing, but with an added symbol in the feature control frame. The M with a circle around it stands for Maximum Material Condition (MMC). For a hole, this is the smallest possible size, and for a shaft, it is the largest. In other words, for mating holes and shafts, MMC is the tolerance condition where fit is most difficult.

If the hole increases in size from MMC, leeway in the position of the hole becomes available. This addition to the positional tolerance zone diameter is defined as the amount that the hole diameter increases over MMC. For a step-by-step example of how bonus tolerancing works, please select "2D Bon Tol Example" from the menu bar. The 3D case is the same except that all of the 2D figures are projected out of the page.

Figure 2.12

As in the 2D case, if no MMC symbol is present, the tolerance is assumed to apply at Regardless of Feature Size (RFS) and there is no bonus tolerance. The Least Material Condition (LMC) also exists, symbolized by a circled L, but we have not found it to be used very often.  A 3D shaft is dimensioned and toleranced in the same way as a 3D hole. For a discussion of the mating of a hole with a shaft, please see 3D Virtual Condition. 3D Virtual Condition

The Virtual Condition of a feature is a concept used to describe the worst-case envelope within which either of two features must lie in order to mate acceptably. For a shaft that fits into a hole, the shaft virtual condition must be smaller than the hole virtual condition. We had discussed this topic in the section 2D Virtual Condition. The concepts and illustrations are the same since tolerance zones and virtual condition boundaries now simply become 3D right circular cylinders that project out of the page.  As we had seen for the 2D case, for an external feature of size, such as a shaft, the virtual condition is equal to the size at MMC plus the size of the tolerance zone. For the shaft in Figure VC1.1, the diameter of the virtual condition is the diameter of the MMC shaft plus the diameter of the position tolerance zone.  

Figure VC1.1

For an internal feature of size, such as a hole, the virtual condition is equal to the size at MMC minus the size of the tolerance zone. For the hole in Figure VC1.2, the diameter of the virtual condition is the diameter of the MMC hole minus the diameter of the position tolerance zone.

Figure VC1.2

Figure VC1.3 shows the shaft and hole virtual conditions superimposed. Since the shaft virtual condition is smaller than the hole virtual condition, the two parts will always mate.

Figure VC1.3

In summary, the way to calculate virtual condition (VC) for a shaft and hole is: SHAFT VC = MMC diameter + Position Tolerance Zone DiameterHOLE VC = MMC diameter - Position Tolerance Zone Diameter

Virtual condition is extremely useful in the design of functional gauges. A functional gauge made to conform to virtual condition will ensure that a part will always mate with its counterpart. See 3D Functional Gauge. 3D Functional Gauge

We have seen how geometric dimensioning and tolerancing to virtual condition ensures part fit. We can use this fact to design a functional gauge that will ensure 100% part fit with minimum good part rejection. We did not cover functional gauges in the 2D section since 2D is more of a theoretical exercise for ease of understanding. In the following example, the objective is for a block with a hole and a block with a shaft to mate with flush sides, as shown in Figure FG3.  Figure FG1 shows a virtual condition gauge for the 3D block that we have been discussing. The block is shown fit to the gauge, with the appropriate datums contacted. The virtual condition is illustrated in VC1.2 in the 3D Virtual Condition section.

Figure FG1

Figure FG2 shows a virtual condition gauge for a 3D shaft block that mates with the hole block. The shaft block is dimensioned very similarly to the hole block. Its datums contact the functional datums of the gauge. The virtual condition is illustrated in VC1.1 in the 3D Virtual Condition section. In the same section, also refer to Figure VC1.3, which shows the shaft and hole virtual condition boundaries superimposed and thus demonstrates guaranteed fit.

Figure FG2

 Figure FG3 shows the final result: Parts that mate perfectly despite rough surfaces, fabrication and inspection at different facilities, and handling-induced changes to dimensions before incoming inspection. This is the true power of GDT: control over the design and manufacturing process which leads to lower costs.  

Figure FG3

In summary, we have seen how GDT functional gauges designed according to virtual condition ensure part fit. Virtual condition is the envelope for worst-case part fit. A part which fits on such a functional gauge is guaranteed to fit to all mating parts. Symbol Glossary

What follows is a short summary of feature control symbols used in GDT. As was mentioned in the introduction, this is not intended to be an exhaustive presentation of GDT, but rather a concise, clear explanation of GDT's most important yet often misunderstood points. Therefore, we do not explain in detail all of the symbols listed below.

In the "Type of Feature" column, "Individual" means that this type of tolerance does not need to be referred to a datum. Individual GDT tolerances are the easiest to understand for a beginning GDT user. "Related" designates a type of tolerance that must be referred to a datum. "Individual or Related" can be of either type, depending on the situation.

SYMBOL

VERBALDESCRIPTION

TYPE of TOLERANCE

TYPE of FEATURE NOTES

Position Location Related Commonly used.

Concentricity Location RelatedDifficult to inspect.

Circular Runout Runout Related

Total Runout Runout RelatedIncludes Circular Runout

Perpendicularity Orientation Related

Parallelism Orientation Related

Angularity Orientation Related

Profile of a Surface Profile Individual or Related

Profile of a Line ProfileIndividual or Related

Flatness Form Individual

Straightness Form Individual

Roundness or Circularity

Form Individual

Cylindricity Form Individual

Parallelism vs. Flatness

One common area of confusion within GDT is between parallelism and flatness. With, parallelism, a reference is made to a datum plane, while flatness is independent of a datum. Figure PF1.1 shows a block on a surface plate whose three points of contact define the -A- datum plane. As specified, the opposite face of the block must lie within two planes which straddle a plane parallel to and 2.0 from -A- and .125 to either side of it.

Figure PF1.1

Figure PF1.2 shows a flatness specification on the same surface. No datum plane is involved. A "best fit" plane (3 point contact) defines the mid-plane, and all surface points must lie between two planes which are parallel to this plane, and .125 to either side of it.

Figure PF1.2

Cylindrical Specifications

One common area of confusion within GDT is the differences between the various ways of specifying how true a cylindrical surface or surface of revolution is: roundness, cylindricity, concentricity, circular runout, and total runout. Let us start with roundness. As shown in Figure CS1, roundness applies to individual circular cross sections of a surface of revolution or of a sphere.  

Figure CS1 : ROUNDNESS

Cylindricity, on the other hand, applies to all cross-sections of a cylindrical surface simultaneously. The surface must lie between the two cylindrical surface which bound the tolerance zone and are determined by a best-fit nominal cylinder. Figure CS2 illustrates cylindricity.  

Figure CS2 : CYLINDRICITY applies to all cross-sectional elements simultaneously.  

It is a common misconception that roundness and cylindricity can be checked by taking diametral measurements (as with a micrometer) or by using an indicator and vee block. A diametral measurement does just what the words imply; it measures the diameter. It does not check the shape of the surface which is what roundness and cylindricity control. Since the roundness or cylindricity tolerance is a radial distance between concentric boundaries, a radial method of checking the surface is necessary. However, rotating a part between centers is not an acceptable method since it relates the part surface to an axis, which technically is a check of another geometric tolerance called runout.  To truly check for the roundness or cylindricity of a surface without regard to the axis of the part, the part must be rotated about the ultra-precision spindle of a specialized roundness measuring machine. A probe contacts the surface and transcribes an enlarged profile of the surface onto a polar graph. The profile is then checked against a clear overlay of concentric circles to determine if it falls within the allowable tolerance zone.  Concentricity is the condition in which the axes of all cross-sectional elements of a surface of revolution are common to the axis of a datum feature. Because the location of the datum axis is difficult to find, it is easier to inspect for cylindricity or runout.  

Figure CS3 : CONCENTRICITY is based upon the datum axis so that it is difficult to

ascertain.

Runout refers to the result of placing a solid of revolution on a spindle such as a lathe, and rotating the part about its central axis while measuring with a dial indicator its surface deviation from perfect roundness. With circular runout, the dial indicator is not moved along the direction of the axis of the part. Circular runout is therefore applied independently at each station along the length of the part as the part is rotated through 360 degrees.  

Figure CS4 : CIRCULAR RUNOUT applies to each cross section individually.

Total runout involves moving the dial indicator along the length of the part while the part is rotated, so that it controls the cumulative variations of circularity, cylindricity, straightness, coaxiality, angularity, taper, and profile.  

Figure CS5 : TOTAL RUNOUT applies to all cross sections simultaneously.  

Acceptance

Unfortunately, GDT has not become universally known as quickly as its originators had envisioned. GDT is a language, and like any language can be intimidating at first. Courses on GDT often quickly present the fundamentals, and then begin with highly complex, specialized applications without making sure that the fundamental

tenets are understood. GDT must be presented at a level that ALL participants in the design, inspection, and manufacturing process are willing to understand. Without this common denominator, GDT will not be an effective communication tool.

Because of this, GDT specifications are often mis-interpereted or left out altogether. This is one unfortunate circumstance which this author attempts to rectify with the current narrative. As computer modelling of mechanical parts continues to increase the complex definition of engineering geometry, and as part tolerances become more and more precise, there is no doubt that GDT will fill the void for the description of 3D geometry for which it was designed. In the current marketplace, companies must produce products at the lowest practicable cost or suffer the consequences. GDT is the only existing way to define mechanical parts so that they fit and function with the widest, cheapest tolerances possible.

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