October 2014 

GMP 9 Page 1 of 2 

GMP 9 

Good Measurement Practice 

for 

Equations for Metallic Tapes 

Table 1. Symbols used in this practice. 

Symbol Description 

L Horizontal straight-line distance 

LS Calibrated length of the tape interval on a flat surface at TS and PS

Ln Designated nominal length of the tape interval 

P Applied tension 

PS Standard tension applied to the tape interval for LS

AE Average cross-sectional area times Young’s Modulus of Elasticity 

T Temperature 

TS Standard temperature of the tape interval for LS, 20 °C (68 °F) 

α Coefficient of thermal expansion of the tape ribbon 

W Average weight per unit length of the tape ribbon 

N Number of equidistant catenary suspensions 

PO Tension of accuracy (Tension of accuracy is defined as that tension which must be 

applied to the tape interval to produce its designated nominal length at the observed 

temperature of the tape.) 

PC Tension of accuracy while the tape is supported in catenary suspensions 

The horizontal straight-line distance, L, of a tape interval can be computed by the following 

equation for an applied tension, P, and temperature, T, when the tape is supported for N, number 

of equidistant catenary suspensions. When both a standard tape and a tape to be calibrated are 

placed in a catenary suspension condition, the horizontal straight line distance for the compared 

intervals for each tape must be calculated. Intervals can be compared only at the supported 

points along the tape length. 

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( * * ) ( ) ( ) 2 L W L N P

L T T

AE

L P P

L L n n

n S

n S

S        (1) 

For simplicity, it is recommended to entirely support the tape on a horizontal flat surface for 

calibration. When the tape is supported entirely on a horizontal flat surface, N = ∞, and the 

general equation is reduced to: 

( ) ( ) n S

n S

S L T T

AE

L P P

L L      (2) 

The distance, L, of the tape interval can be set to the designated nominal length, Ln, for 

determining the tension of accuracy, PO, while the tape is supported on a flat surface, by writing 

equation (2) as follows: 
October 2014 

GMP 9 Page 2 of 2 

( ) ( ) n S

n O S

n S L T T

AE

L P P

L L      (3) 

from which the following equations are developed: 

( ) ( )

S

n

n S

O S AE T T

L

AE L L

P P      (4) 

or 

( ) ( )

S

n

n S

S O AE T T

L

AE L L P P      (5) 

Substituting equation (5) for PS in the general equation (1), we have 

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( ) ( * * )

2

n O n n C

n

L W L N P

AE

L P P

L L     (6) 

The distance, L, of the tape interval again can be set to the designated nominal length, Ln, for 

determining the tension of accuracy, PC, while the tape is supported in catenary suspensions, by 

writing this equation as follows 

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( ) ( * * )

2

n C O n n C

n n

L W L N P

AE

L P P

L L     (7) 

from which: 

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( * ) ( )

2

2 AE W L N P P P n

C C  O  (8) 

or 

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( * ) ( ) ( ) 2

2 AE W L N AE T T

L

AE L L P P P n

S

n

n S

C C S  



 





      (9) 

The value of PC can be solved by first determining the right side of the equal signs in equations 

(8) or (9), then substituting various values for PC until the left side approaches the right side 

within the desired limits. If the value is greater than the right side, reduce the value of PC.