GD&T Positional Tolerances

The GD&T positional tolerance defines a tolerance zone within which the centre of the hole must be located. The hole is located with respect to one or more datums.

In most cases, this is exactly what you want to specify, so positional tolerances ought to be the preferred method.

There are several ways to apply tolerances to holes and to their positions.

Positional Tolerances Using the Bolt and Screw Models

Figure 8: Bolt Located by a Positional Tolerance

See Figure 8 . $G$ is the positional tolerance to be called up on the drawing for a bolted assembly. $c$ is the diametral clearance.

G_b = 2 \times {\Delta_b}= 2 \times \left( \frac{c}{2} \right)

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

\fbox{$ \displaystyle G_b = c $}
\end{displaymath} (11)

For a screwed assembly...

\dots G_s = 2 \times {\Delta_s}= 2 \times \left( \frac{c}{4} \right)

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

\fbox{$ \displaystyle G_s = \frac{c}{2} $}
\end{displaymath} (12)

Positional Tolerances from First Principles

Let's ignore the bolt and screw models preceding.

We want to install a fastener, and we want the clearance holes to clear it. In the following analyses, we assume that the parts are located by datums A, B and C which are external to the hole pattern.

Bolt from First Principles

Figure 9: Bolted Connection with maximum position error

We locate our bolt at the exact nominal position.

Located exactly at nominal position, an infinitesimally larger clearance hole will clear the bolt. As the hole gets larger, it can shift some distance off nominal and still clear the bolt, as shown in Figure 9 . Note that the geometric tolerance $G$ applies to the hole, only.

Figure: Specification of Figure 9 's Geometry

Figure 10 shows the dimension specification we want to use on each part that requires a clearance hole. The zero positional tolerance applies only at MMC, the minimum clearance diameter. In the LMC case, you have the maximum sized hole, shifted the maximum clearance off to one side. The maximum diameter allows for the off-centre error as well as the error in the hole diameter.

\fbox{$ MIN_{BOLT} = D_{BOLT} + \mbox{minimum clearance} $}
\end{displaymath} (13)

The hole must be larger than the bolt.

For our maximum diameter, we must allow for a reasonable positional tolerance $G$ plus a reasonable tolerance on the hole diameter which would be $\pm t$.

\fbox{$ MAX_{BOLT} = MIN_{BOLT} + G + 2t $}
\end{displaymath} (14)

Screw From First Principles

Figure 11: Screwed Connection with maximum position error

See Figure 11 . For simplicity, we assume that the positional tolerance $G$ is the same for the tapped hole and the clearance hole.

Figure 12: Tapped hole and clearance hole with positional tolerances

Our two parts require the dimension specifications shown in Figure 12 . The tapped hole shown is metric, with a major diameter of $D_{SCREW}$ and a pitch $P$ which is not used on these calculations.

The screw is potentially able to occupy a space of $D_{SCREW}+G$, as shown on Figure 11 . The clearance hole must clear this.

\fbox{$ MIN_{SCREW} = D_{SCREW} + G $}
\end{displaymath} (15)

The positional tolerance of the tapped hole controls from the part's surface to the bottom of the tapped hole. Your clearance hole must clear whatever sticks out of the tapped hole. If your clearance hole goes through a thick plate, you should add a projected tolerance zone to your feature control frame, or at least, increase $MIN_{SCREW}$ a bit.

Specifying MMC on a tapped hole makes no sense to me. Tapped holes are self-centreing, so there is no bonus tolerance!

For the clearance hole, again, we add a reasonable positional tolerance $G$, and the clearance hole tolerance $\pm t$ to get...

\fbox{$ MAX_{SCREW} = MIN_{SCREW} + G + 2t $}
\end{displaymath} (16)

It is not necessary for the positional tolerance $G$ to be identical for the tapped hole and the clearance hole. If the tapped hole is in a machined part and the clearance is in a sheet metal part or a weldment, an accurate $G_{TAP}$ will allow for a sloppier $G_{CLEAR}$. Or, vice versa!

Checking Zero Positional Tolerances

If you are checking a drawing dimensioned like Figure 10 , you can verify that the tolerance is fabricatable.

G + 2t = MAX - MIN

Obviously, maximally tight values for $G$ and $\pm t$ should not exceed the difference between $MAX$ and $MIN$.

Calculations from ASME Y14.5M-1994 Appendix B

ASME Y14.5M-1994 proposes the fixed fastener case, equivalent to my bolts, and the floating fastener case, equivalent to my screws.

$D$ = minimum depth of thread or minimum thickness of part with restrained or fixed fastener
$H$ = maximum diameter of fastener (MMC limit)
$F$ = minimum diameter of clearance hole (MMC limit)
$P$ = maximum thickness of part with clearance hole, or maximum projection of fastener, such as a stud.
$T$ = positional tolerance diameter

The floating fastener case...

\begin{displaymath}H = F + T\end{displaymath}


\begin{displaymath}T = H - F \end{displaymath}

The fixed fastener case...

\begin{displaymath}H = F + 2T \end{displaymath}


\begin{displaymath}T = \frac{H - F}{2} \end{displaymath}

An interesting problem came up on Eng-Tips forums, We are assembling two parts, one of which was designed and manufactured outside. We control the dimensions and tolerances only of the other part. The device is assembled using screws and nuts, so the floating fastener case applies. We can modify the equation for two sets of tolerances as follows...

\begin{displaymath}H_1 + H_2 = 2F + T_1 + T_2 \end{displaymath}

The manufactured part has a clearance hole $H_1=0.104''$ and they have specified an orthogonal tolerance of $\pm 0.005''$. This is equivalent to a geometric tolerance of $T_1=0.014''$.

Let's set $T_2=0$, and solve for $H_2$.

H_2 & = & T_2 + T_1 + 2F - H_1 \\
& = & 0 + 0.014'' + 2 \times 0.086'' - 0.104'' \\
& = & 0.082'' \\

The calculated clearance hole is smaller than the screw!

Let's try again, setting $H_2=.086''$, and solving for $T_2$.

T_2 & = & H_1 + H_2 - 2F - T_1 \\
& = & 0.104'' + 0.086'' - 2 \times 0.086'' - 0.014'' \\
& = & 0.004'' \\

Now the result makes sense. If you provide a zero clearance hole, you may allow a 0.004" positional tolerance. This violates our standard model in which the bolt is located at the exact nominal position, but it is a perfectly functional assembly.

When you solve this problem, solve for the tolerance, not the hole diameter.