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SOP 23 

Recommended Standard Operations Procedure for 

Calibrations of PI Tapes 

Bench Method 

1. Introduction 

1.1 Purpose 

PI tapes are circumferential measuring tapes that are wrapped around a circular or 

cylindrical object and when read, indicate the diameter of the object. The tapes 

are manufactured for various diameter ranges and for either outside diameters 

(OD) or inside diameters (ID). The test method described here provides a 

procedure to calibrate such tapes to four decimal places in inches (diameter). A 

calibrated length bench (transfer standard, calibrated using SOP 11) is used as the 

standard. 

1.2 Prerequisites 

1.2.1 Valid calibration certificates with appropriate values and uncertainties 

must be available for all of the standards used in the calibration. All 

standards must have demonstrated metrological traceability to the 

international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST. 

1.2.2. The ocular microscope used in measuring differences in lengths must be in 

good operating condition and must be equipped with a graduated reticle 

having established traceability. 

1.2.3. The operator must be trained and experienced in precision measuring 

techniques with specific training in GMP 2, GMP 8, GMP 9, SOP 11, SOP 

23, and SOP 29. 

1.2.4. Laboratory facilities must comply with following minimum conditions to 

meet the expected uncertainty possible with this procedure. Equilibration 

of length bench and tapes requires environmental stability of the 

laboratory within the stated limits for a minimum of 24 hours before a 

calibration. 

Table 1. Environmental conditions. 

Temperature Requirements During a Calibration Relative Humidity 

(%) 

Lower and upper limits: 18 °C to 22 °C 

Maximum changes: < ± 1 °C / 24 h and ± 0.5 °C / 1 h 40 to 60 ± 10 / 4 h 
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2. Methodology 

2.1. Scope, Precision, Accuracy 

The precision and accuracy attainable depend upon the care exercised in aligning 

the tape on the length bench, and the skill acquired in the use of an optical 

micrometer to measure scale differences. 

2.2 Physical Characteristics 

2.2.1. PI tapes are manufactured with corrections built into the tape where they 

will indicate the proper diameter without using extensive mathematical 

corrections. These corrections will have to be taken into consideration 

when calibrating a PI tape on a flat horizontal surface. 

OD PI tapes are referenced to 20 °C with 2.25 kg (5 lb) tension (within ± 

0.5 percent of the load, i.e., Class F mass standards are more than 

adequate.) applied. There is concern about the proper tension to apply 

to an ID PI tape since they are normally used under some degree of 

compression. Most tapes are made of 1095 spring steel, which has a 

coefficient of thermal expansion of 0.00000633 / °F. 

Figure 1. Schematic drawing of Pi tape 

 As the tape is wrapped around the circumference of an object, the reading 

portion of the tape and the vernier come into alignment. The smallest 

graduation on the main body equals 0.5 mm (0.025 in). The line on the 
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vernier scale which comes closest to lining up with a graduation on the 

main body is the value to be added to the reading. The vernier allows 

readings to be made to the nearest 0.001 in, as shown in Figure 2. 

 

 Figure 2. Reading the Vernier scale on a Pi tape. 

2.3. Fundamental Considerations 

2.3.1. The relationship of circumference, C, diameter, D, and pi () is C = (D), 

D = C/. See Figure 3 where  = 3.141592654 

 

 Figure 3. Circumference, diameter, pi ().

2.3.2. The thickness of the tape will affect the measurements in two locations, as 

shown in Figure 4. L1 is the outside portion of the tape, which becomes 

longer when the tape is bent. L2 is the center of the tape (neutral axis). It 

remains the same length when the tape is bent. L3 is the inside portion of 

the tape and it compresses and becomes shorter when the tape is bent. 

 Figure 4. Side view of tape.
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2.3.3 Another point that must be taken into consideration is that the tape was 

tested for accuracy on a flat horizontal surface but now it is curved around 

an object. The tape becomes temporarily deformed but the amount of 

distortion is predictable using the modulus of elasticity. As shown in the 

diagram (Figure 4.), the outside portion of the tape stretches and the inside 

compresses but there is a point midway through the thickness of the tape 

which does not change in length and is called the neutral axis. 

For all homogeneous beams with the same modulus of elasticity in tension 

and compression, the neutral axis passes through the center of the section. 

If the tape has a thickness of 0.010 inch, the distance from the neutral axis 

to the surface of the object is 0.005 inch. Therefore, the reading we 

obtained is not the distance on the outside surface of the tape but is 

actually the distance along the neutral axis of the tape. When we calculate 

a diameter, this is actually the diameter of the neutral axis of the tape. 

Since this thickness affects the diameter in two locations, the amount of 

correction that must be taken into account is 0.010 inches. 

2.4. Summary 

The tape to be calibrated is laid over the bench scale and tension is applied to 

ensure that it lies flat on the bench. Differences between the graduation on the 

tape and that of the bench scale are measured using an ocular micrometer. The 

temperature of the tape is observed and corrections applied to the reference 

temperature of 20 °C. Tapes longer than the length bench may be calibrated in 

segments, using the last calibrated graduation as the zero graduation mark for the 

succeeding segments. PI tape graduations represent object diameters and may not 

directly coincide with length bench graduations. 

2.5. Apparatus/Equipment Required 

2.5.1. Calibrated thermometers with sufficiently small resolution, stability, and 

uncertainty, capable of indicating temperatures in the range of 15 °C to 

30 °C, and accurate to ± 0.5 °C. 

2.5.2. Calibrated 5 meter (16 foot) length bench. 

2.5.3. Microscope with graduated reticle having graduations spaced at intervals 

no greater than 0.002 inch. 

2.5.4. Clamps, fabric tape in lengths required to connect tension weights to the 

tape being calibrated, and weights to apply an appropriate tension to the 

tape under calibration. (See "Tension Specifications", in Appendix C.3 in 

SOP 11). 
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2.6. Symbols 

Table 2. Symbols used in this procedure. 

Symbol Description 

S Standard 

X Unknown 

d0i Initial zero difference 

d0f Final zero difference 

X0L Left edge of zero on unknown 

X0R Right edge of zero on unknown 

S0L Left edge of zero on standard 

S0R Right edge of zero on standard 

Xm Center of graduation of unknown 

Sm Center of graduation of standard 

d Difference of interval between X and S

t Average of initial and final temperatures 

tcorr Temperature correction 

Ln Nominal length of tape interval under test in inches 

LX Calibrated length of tape interval under test in inches 

α Linear coefficient of thermal expansion for the standard bench (0.00001063 / oC)

β Linear coefficient of thermal expansion for the tape (0.00001160 / oC) 

Ld Diameter length of interval 

π 3.141592654 

2.7. Procedure 

2.7.1. Inspect the tape to ensure that no kinks, dents, or other damage are present 

which will affect the accuracy of the tape. 

2.7.2. Clean the tape by first wiping with a soft cloth, and then with a soft cloth 

saturated with alcohol to remove the protective oil film. (See Appendix C, 

Section C.1 for SOP 11). 

2.7.3. Place the tape, under the clamp, at the zero end of the bench, and adjust so 

the starting mark on the tape is near the zero graduation on the length 

bench. 

2.7.4. Lay the tape flat on the bench. The tape should extend well beyond the 

end roller of the bench to permit tension to be applied. Slide one end of a 

fabric strap onto another tape clamp. Fasten this tape clamp to the tape on 

the portion that extends below the end roller. Hang the tension weight 

from the bottom of the fabric strap. Check to see that the tape is lying 

straight on the bench and parallel to the bench scale. Adjust, if necessary. 

Apply tension using a weight of 2.25 kg (5 lb) unless other tension is 

specified. (See Appendix C.3 in SOP 11.) When a tape is shorter than the 
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5 m length bench, fabric strap will be be used to effectively lengthen the 

tape so that the tension weight is properly suspended below the rollers of 

the length bench. 

2.7.5. Lay two thermometers (see 2.4.1.) on the bench at intervals of one-third 

and two-thirds of the length of the bench to determine its temperature at 

the time of test. 

2.7.6. Adjust the tape clamp on the zero end of the bench so that the tape zero 

graduation coincides with the center of the zero graduation of the bench. 

2.7.7. Check all alignments and coincidence of zero graduations before 

proceeding with the calibration. Use the lateral adjustments of the left end 

of the bench to facilitate alignment. Caution! Take care that the tape is not 

touched or disturbed during the following sequence of measurements. 

Record all observations on a suitable data sheet. (See SOP 11, Appendix 

A.) 

2.7.8. Record the temperatures indicated by the two thermometers. 

2.7.9. Place the ocular microscope on the bench in the vicinity of the zero 

position and align it so that its scale is parallel to the tape under test. (See 

GMP No. 2 for instructions on how to make readings.) 

2.7.9.1.Observe readings of left and right sides of zero graduation of tape 

and record to the nearest 0.001 inch. The mean of these values 

will be used to determine the center of the tape graduation. 

2.7.9.2.Observe readings of left and right sides of zero graduation of 

bench and record to the nearest 0.001 inch. The mean of these 

values will be used to determine the center of the bench 

graduation. 

2.7.10. Move the ocular microscope successively to each graduation that is to be 

calibrated and record readings similarly as in 2.7.9.1 and 2.7.9.2. 

2.7.11. Return the ocular microscope to the zero graduation and repeat readings to 

verify that the tape has not moved. Accept all previous data if the initial 

zero reading does not disagree with the final zero reading by more than 

0.001 inch; otherwise, discard previous data and repeat entire sequence of 

readings until a satisfactory set is obtained. 

2.7.12. Loosen the tape by removing the tension weights, move the tape, hang the 

tension weights back on the fabric strap, and realign the zero marks on the 

tape and bench to coincidence. 
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2.7.13. Repeat 2.7.9. thru 2.7.12. for every 15 foot (5 meter) section of the tape 

requiring calibration. This will require repositioning the tape, aligning the 

last measured interval graduation on the tape with the zero graduation of 

the bench. 

2.7.14. Make a second set of measurements as directed in 2.7.9 through 2.7.13. 

2.7.15. After the final measurement is taken and accepted, record the temperatures 

indicated by the two thermometers. 

2.7.16. After all measurements are completed, apply a thin film of oil to the tape. 

3. Calculations 

3.1. Calculate the Initial Zero Difference, d0i (2.6.9.) 

 2

( ) 0 0 0 0

0

L R L R

i

X X S S d     (1) 

 3.2. Calculate the Final Zero Difference, d0f (2.6.11.) 

2

( ) 0 0 0 0

0

L R L R

f

X X S S d     (2) 

3.3. Calculate the center of graduation for unknown, Xm, and standard, Sm, for each set of 

measurements and each scale interval. 

2

( ) L R

m

X X X   (3) 

2

( ) L R

m

S S S   (4) 

3.4. Calculate the difference, d, between X and S for each scale interval. 

m m d  X  S (5) 

3.5. Obtain the correction to the standard, CS, for the measured interval from the 

calibration certificate for the length bench scale. 

 3.6. Calculate the temperature correction, tcorr. 

t L     t  20 (  ) corr n (6)

3.7. Calculate a correction, CX, for each trial and each scale interval. 
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X S corr C  d  C  t (7) 

3.8. Calculate and report the mean,CX , of the two corrections for each interval. 

3.9. Calculate the length of tape under the 10 pound load. 

LX  Ln  CX (8)

3.10. As appropriate, convert the flat length of the tape to an outside diameter (OD) 

measurement using the following formula: 

 (tape thickness in inches) L L X

OD   

 (9) 

3.11. As appropriate, convert the flat length of the tape to an inside diameter (ID) 

measurement using the following formula: 

 (tape thickness in inches) L L X

ID  



 (9) 

4. Measurement Assurance 

4.1. Duplicate the process with a suitable check standard or have a suitable range of 

check standards for the laboratory. See NISTIR 7383 SOP 17, SOP 20 and 

NISTIR 6969 SOP 30. Plot the check standard length and verify it is within 

established limits OR a t-test may be incorporated to check the observed value 

against an accepted value. The mean of the check standard observations is used to 

evaluate bias and drift over time. Check standard observations are used to 

calculate the standard deviation of the measurement process which contributes to 

the Type A uncertainty components. 

4.2 If a standard deviation chart is used for measurement assurance, the standard 

deviation of each combination of Trial 1 and Trial 2 is calculated and the pooled 

(or average) standard deviation is used as the estimate of variability in the 

measurement process. Note: the pooled or average standard deviation over time 

will reflect varying conditions of test items that are submitted to the laboratory. A 

standard deviation chart will be needed for each interval calibrated (at least 

initially) so that the variability resulting from transfers will be measured. 

5. Assignment of Uncertainty 

5.1. The limits of expanded uncertainty, U, include estimates of the standard 

uncertainty of the length standards used, us, estimates of the standard deviation of 

the measurement process, sp, and estimates of the effect of other components 

associated with this procedure, uo. These estimates should be combined using the 

root-sum-squared method (RSS), and the expanded uncertainty, U, reported with 

a coverage factor to be determined based on degrees of freedom, which if large 
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enough will be 2, (k = 2), to give an approximate 95 percent level of confidence. 

See NISTIR 6969, SOP 29 (Standard Operating Procedure for the Assignment of 

Uncertainty) for the complete standard operating procedure for calculating the 

uncertainty. 

5.1.1 The expanded uncertainty for the standard, U, is obtained from the 

calibration report. The combined standard uncertainty, uc, is used and not 

the expanded uncertainty, U, therefore the reported uncertainty for the 

standard will usually need to be divided by the coverage factor k. 

5.1.2. The standard deviation of the measurement process, sp, is taken from a 

control chart for a check standard or standard deviation charts. (See SOP 

17, SOP 20, and SOP 30) 

5.1.3. Uncertainty associated with bias, ud. Any noted bias that has been 

determined through analysis of control charts and round robin data must 

be less than limits provided in SOP 29 and included if corrective action is 

not taken. See SOP 29 for additional details 

5.1.4. Uncertainty associated with temperature correction includes values for the 

linear coefficients of expansion for the Standard and Unknown, the 

accuracy of temperature measurements, and factors associated with 

potential gradients in measuring the temperature on the length bench and 

tape. 

5.1.5. Other standard uncertainties usually included at this calibration level 

include uncertainties associated with the ability to read the graduated 

reticle, only part of which is included in the process variability due to 

parallax and visual capabilities, and uncertainties associated with the 

graduations of the reticle. 
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 Table 3. Example uncertainty budget table. 

Component Description Reference 

us Standard uncertainty for standards Calibration report, divide by k 

sp Standard uncertainty for the process Measurement assurance process; 

range charts 

ugr Standard uncertainty for graduated reticle Must be assessed experimentally 

or from a calibration certificate 

utc Standard uncertainty for temperature 

correction 

Calibration certificate 

 ut Standard uncertainty for temperature Handbook 143 accuracy 

guideline, 0.1 C 

 ulce (,) Standard uncertainty for linear coefficient 

of expansion 

5 % to 10 % of the coefficient of 

expansion value 

utw Standard uncertainty for tension weights Calibration certificate 

ud Standard uncertainty for disparity due to 

drift/bias 

Rectangular distribution and 

reasons, 0.577 d, 0.29 d; 

SOP 29 (NISTIR 6969) 

ures Standard uncertainty due to resetting of 

the tape 

Must be assessed experimentally 

uo Standard uncertainty for other factors 

6. Report 

Report results as described in SOP No. 1 Preparation of Calibration Certificates. 

 
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Appendix A 

Bench Method Data Sheet 

Date Environmental parameters 

Metrologist Before After Unc/ability to measure 

Test No. Temperature °C °C 

sp in Pressure mmHg mmHg 

df Humidity % % 

Based on NISTIR 6969, SOP 29, Appendix A at 95.45 % probability distribution: k factor 

 ID Range Linear coefficient of thermal expansion 

S Bench 16 ft (5 m) α 0.000 010 63 /°C 

X Tape β 0.000 011 60 /°C 

Tension lb Support 

Initial temperatures Final temperatures 

t1 °C t3 °C Average temperature °C 

t2 °C t4 °C 

Interval, ft Trial Initial Zero Graduations 

Bench Tape Tape Bench 

0 0 

1 X0L X0R S0L S0R

 

Initial zero, difference between X and S, d0i in 

2 X0L X0R S0L S0R 

 

Initial zero, difference between X and S, d0i in 

Intervals after zero 

Tape Bench 

 

 X1L X1R Xm S1L S1R Sm d CS tcorr CX Range 

1 

2 

Average CX in 

 

 X1L X1R Xm S1L S1R Sm d CS tcorr CX Range 

1 

2 

Average CX in 

Final Zero Graduations 

Tape Bench 

0 0 

1 

X0L X0R S0L S0R

 

Final zero, difference between X and S, d0f in 

Difference between Initial Zero and Final Zero, d0i - d0f in 

| d0i - d0f | ≤ 0.001 in? If YES, accept previous data. 

2 

X0L X0R S0L S0R 

 

Final zero, difference between X and S, d0f in 

 Difference between Initial Zero and Final Zero, d0i - d0f in 

| d0i - d0f | ≤ 0.001 in? If YES, accept previous data. 

t L     t  20 (  ) corr n X S corr C  d  C  t LX  Ln  CX