Levelling & Contouring | Geomatics Engineering (Surveying) - Civil Engineering (CE) PDF Download

 Page 1


SURVEYING
FORMULAS
UNITS OF MEASUREMENT
Units of measurement used in past and present surveys are
For construction work: feet, inches, fractions of inches (m, mm)
For most surveys: feet, tenths, hundredths, thousandths (m, mm)
For National Geodetic Survey (NGS) control surveys: meters, 0.1, 0.01, 0.001 m
The most-used equivalents are
1 meter  39.37 in (exactly)  3.2808 ft
1 rod  1 pole  1 perch  ft (5.029 m)
1 engineer’s chain  100 ft  100 links (30.48 m)
1 Gunter’s chain  66 ft (20.11 m)  100 Gunter’s links (lk)  4 rods  mi
(0.020 km)
1 acre  100,000 sq (Gunter’s) links  43,560 ft
2
 160 rods
2
 10 sq
(Gunter’s) chains  4046.87 m
2
 0.4047 ha
1 rood  acre (1011.5 m
2
)  40 rods
2
(also local unit  to 8 yd) 
(5.029 to 7.315 m)
1 ha  10,000 m
2
 107,639.10 ft
2
 2.471 acres
1 arpent  about 0.85 acre, or length of side of 1 square arpent (varies) (about
3439.1 m
2
)
1 statute mi  5280 ft  1609.35 m
1 mi
2
 640 acres (258.94 ha)
1 nautical mi (U.S.)  6080.27 ft  1853.248 m
1 fathom  6 ft (1.829 m)
1 cubit  18 in (0.457 m)
1 vara  33 in (0.838 m) (Calif.), in (0.851 m) (Texas), varies
1 degree  circle  60 min  3600 s  0.01745 rad
sin 1 0.01745241
1

360
33
1

3
5
1

2
3

4
1

80
16
1

2
Page 2


SURVEYING
FORMULAS
UNITS OF MEASUREMENT
Units of measurement used in past and present surveys are
For construction work: feet, inches, fractions of inches (m, mm)
For most surveys: feet, tenths, hundredths, thousandths (m, mm)
For National Geodetic Survey (NGS) control surveys: meters, 0.1, 0.01, 0.001 m
The most-used equivalents are
1 meter  39.37 in (exactly)  3.2808 ft
1 rod  1 pole  1 perch  ft (5.029 m)
1 engineer’s chain  100 ft  100 links (30.48 m)
1 Gunter’s chain  66 ft (20.11 m)  100 Gunter’s links (lk)  4 rods  mi
(0.020 km)
1 acre  100,000 sq (Gunter’s) links  43,560 ft
2
 160 rods
2
 10 sq
(Gunter’s) chains  4046.87 m
2
 0.4047 ha
1 rood  acre (1011.5 m
2
)  40 rods
2
(also local unit  to 8 yd) 
(5.029 to 7.315 m)
1 ha  10,000 m
2
 107,639.10 ft
2
 2.471 acres
1 arpent  about 0.85 acre, or length of side of 1 square arpent (varies) (about
3439.1 m
2
)
1 statute mi  5280 ft  1609.35 m
1 mi
2
 640 acres (258.94 ha)
1 nautical mi (U.S.)  6080.27 ft  1853.248 m
1 fathom  6 ft (1.829 m)
1 cubit  18 in (0.457 m)
1 vara  33 in (0.838 m) (Calif.), in (0.851 m) (Texas), varies
1 degree  circle  60 min  3600 s  0.01745 rad
sin 1 0.01745241
1

360
33
1

3
5
1

2
3

4
1

80
16
1

2
1 rad  57 17

44.8

or about 57.30
1 grad (grade)  circle  quadrant  100 centesimal min  10
4
cen-
tesimals (French)
1 mil  circle  0.05625
1 military pace (milpace)  ft (0.762 m)
THEORY OF ERRORS
When a number of surveying measurements of the same quantity have been
made, they must be analyzed on the basis of probability and the theory of
errors. After all systematic (cumulative) errors and mistakes have been elimi-
nated, random (compensating) errors are investigated to determine the most
probable value (mean) and other critical values. Formulas determined from
statistical theory and the normal, or Gaussian, bell-shaped probability distrib-
ution curve, for the most common of these values follow.
Standard deviation of a series of observations is
(7.1)
where d  residual (difference from mean) of single observation and n  num-
ber of observations.
The probable error of a single observation is 
(7.2)
(The probability that an error within this range will occur is 0.50.)
The probability that an error will lie between two values is given by the
ratio of the area of the probability curve included between the values to the total
area. Inasmuch as the area under the entire probability curve is unity, there is a
100 percent probability that all measurements will lie within the range of the
curve.
The area of the curve between
s
is 0.683; that is, there is a 68.3 percent
probability of an error between 
s
in a single measurement. This error range
is also called the one-sigma or 68.3 percent confidence level. The area of the
curve between  2
s
is 0.955. Thus, there is a 95.5 percent probability of an
error between  2
s
and  2
s
that represents the 95.5 percent error (two-
sigma or 95.5 percent confidence level). Similarly,  3
s
is referred to as the
99.7 percent error (three-sigma or 99.7 percent confidence level). For practical
purposes, a maximum tolerable level often is assumed to be the 99.9 percent
error. Table 7.1 indicates the probability of occurrence of larger errors in a sin-
gle measurement.
The probable error of the combined effects of accidental errors from differ-
ent causes is
(7.3) E
sum

2
E
2
1
 E
2
2
 E
2
3
 
PE
s
0.6745
s

s

B

d
2
n  1
2
1

2
1

6400
1

100
1

400
Page 3


SURVEYING
FORMULAS
UNITS OF MEASUREMENT
Units of measurement used in past and present surveys are
For construction work: feet, inches, fractions of inches (m, mm)
For most surveys: feet, tenths, hundredths, thousandths (m, mm)
For National Geodetic Survey (NGS) control surveys: meters, 0.1, 0.01, 0.001 m
The most-used equivalents are
1 meter  39.37 in (exactly)  3.2808 ft
1 rod  1 pole  1 perch  ft (5.029 m)
1 engineer’s chain  100 ft  100 links (30.48 m)
1 Gunter’s chain  66 ft (20.11 m)  100 Gunter’s links (lk)  4 rods  mi
(0.020 km)
1 acre  100,000 sq (Gunter’s) links  43,560 ft
2
 160 rods
2
 10 sq
(Gunter’s) chains  4046.87 m
2
 0.4047 ha
1 rood  acre (1011.5 m
2
)  40 rods
2
(also local unit  to 8 yd) 
(5.029 to 7.315 m)
1 ha  10,000 m
2
 107,639.10 ft
2
 2.471 acres
1 arpent  about 0.85 acre, or length of side of 1 square arpent (varies) (about
3439.1 m
2
)
1 statute mi  5280 ft  1609.35 m
1 mi
2
 640 acres (258.94 ha)
1 nautical mi (U.S.)  6080.27 ft  1853.248 m
1 fathom  6 ft (1.829 m)
1 cubit  18 in (0.457 m)
1 vara  33 in (0.838 m) (Calif.), in (0.851 m) (Texas), varies
1 degree  circle  60 min  3600 s  0.01745 rad
sin 1 0.01745241
1

360
33
1

3
5
1

2
3

4
1

80
16
1

2
1 rad  57 17

44.8

or about 57.30
1 grad (grade)  circle  quadrant  100 centesimal min  10
4
cen-
tesimals (French)
1 mil  circle  0.05625
1 military pace (milpace)  ft (0.762 m)
THEORY OF ERRORS
When a number of surveying measurements of the same quantity have been
made, they must be analyzed on the basis of probability and the theory of
errors. After all systematic (cumulative) errors and mistakes have been elimi-
nated, random (compensating) errors are investigated to determine the most
probable value (mean) and other critical values. Formulas determined from
statistical theory and the normal, or Gaussian, bell-shaped probability distrib-
ution curve, for the most common of these values follow.
Standard deviation of a series of observations is
(7.1)
where d  residual (difference from mean) of single observation and n  num-
ber of observations.
The probable error of a single observation is 
(7.2)
(The probability that an error within this range will occur is 0.50.)
The probability that an error will lie between two values is given by the
ratio of the area of the probability curve included between the values to the total
area. Inasmuch as the area under the entire probability curve is unity, there is a
100 percent probability that all measurements will lie within the range of the
curve.
The area of the curve between
s
is 0.683; that is, there is a 68.3 percent
probability of an error between 
s
in a single measurement. This error range
is also called the one-sigma or 68.3 percent confidence level. The area of the
curve between  2
s
is 0.955. Thus, there is a 95.5 percent probability of an
error between  2
s
and  2
s
that represents the 95.5 percent error (two-
sigma or 95.5 percent confidence level). Similarly,  3
s
is referred to as the
99.7 percent error (three-sigma or 99.7 percent confidence level). For practical
purposes, a maximum tolerable level often is assumed to be the 99.9 percent
error. Table 7.1 indicates the probability of occurrence of larger errors in a sin-
gle measurement.
The probable error of the combined effects of accidental errors from differ-
ent causes is
(7.3) E
sum

2
E
2
1
 E
2
2
 E
2
3
 
PE
s
0.6745
s

s

B

d
2
n  1
2
1

2
1

6400
1

100
1

400
where E
1
, E
2
, E
3
. . . are probable errors of the separate measurements.
Error of the mean is 
(7.4)
where E
s
 specified error of a single measurement.
Probable error of the mean is 
(7.5)
MEASUREMENT OF DISTANCE WITH TAPES
Reasonable precisions for different methods of measuring distances are
Pacing (ordinary terrain): to 
Taping (ordinary steel tape): to (Results can be improved by use of
tension apparatus, transit alignment, leveling.)
Baseline (invar tape): to 
Stadia: to (with special procedures)
Subtense bar: to (for short distances, with a 1-s theodolite, averag-
ing angles taken at both ends)
Electronic distance measurement (EDM) devices have been in use since the
middle of the twentieth century and have now largely replaced steel tape mea-
surements on large projects. The continued development, and the resulting drop
in prices, are making their use widespread. A knowledge of steel-taping errors
and corrections remains important, however, because use of earlier survey data
requires a knowledge of how the measurements were made, common sources for
errors, and corrections that were typically required.
1

7000
1

1000
1

500
1

300
1

1,000,000
1

50,000
1

10,000
1

1000
1

100
1

50
PE
m

PE
s
n
0.6745
B

d
2
n(n  1)
E
m

E
sum
n

E
s
n
n

E
s
n
TABLE 7.1 Probability of Error in a Single Measurement
Probability
Confidence of larger
Error level, % error 
Probable (0.6745
s
) 50 1 in 2
Standard deviation (
s
) 68.3 1 in 3
90% (1.6449
s
) 90 1 in 10
2
s
or 95.5% 95.5 1 in 20
3
s
or 97.7% 99.7 1 in 370
Maximum (3.29
s
) 99.9 1 in 1000
Page 4


SURVEYING
FORMULAS
UNITS OF MEASUREMENT
Units of measurement used in past and present surveys are
For construction work: feet, inches, fractions of inches (m, mm)
For most surveys: feet, tenths, hundredths, thousandths (m, mm)
For National Geodetic Survey (NGS) control surveys: meters, 0.1, 0.01, 0.001 m
The most-used equivalents are
1 meter  39.37 in (exactly)  3.2808 ft
1 rod  1 pole  1 perch  ft (5.029 m)
1 engineer’s chain  100 ft  100 links (30.48 m)
1 Gunter’s chain  66 ft (20.11 m)  100 Gunter’s links (lk)  4 rods  mi
(0.020 km)
1 acre  100,000 sq (Gunter’s) links  43,560 ft
2
 160 rods
2
 10 sq
(Gunter’s) chains  4046.87 m
2
 0.4047 ha
1 rood  acre (1011.5 m
2
)  40 rods
2
(also local unit  to 8 yd) 
(5.029 to 7.315 m)
1 ha  10,000 m
2
 107,639.10 ft
2
 2.471 acres
1 arpent  about 0.85 acre, or length of side of 1 square arpent (varies) (about
3439.1 m
2
)
1 statute mi  5280 ft  1609.35 m
1 mi
2
 640 acres (258.94 ha)
1 nautical mi (U.S.)  6080.27 ft  1853.248 m
1 fathom  6 ft (1.829 m)
1 cubit  18 in (0.457 m)
1 vara  33 in (0.838 m) (Calif.), in (0.851 m) (Texas), varies
1 degree  circle  60 min  3600 s  0.01745 rad
sin 1 0.01745241
1

360
33
1

3
5
1

2
3

4
1

80
16
1

2
1 rad  57 17

44.8

or about 57.30
1 grad (grade)  circle  quadrant  100 centesimal min  10
4
cen-
tesimals (French)
1 mil  circle  0.05625
1 military pace (milpace)  ft (0.762 m)
THEORY OF ERRORS
When a number of surveying measurements of the same quantity have been
made, they must be analyzed on the basis of probability and the theory of
errors. After all systematic (cumulative) errors and mistakes have been elimi-
nated, random (compensating) errors are investigated to determine the most
probable value (mean) and other critical values. Formulas determined from
statistical theory and the normal, or Gaussian, bell-shaped probability distrib-
ution curve, for the most common of these values follow.
Standard deviation of a series of observations is
(7.1)
where d  residual (difference from mean) of single observation and n  num-
ber of observations.
The probable error of a single observation is 
(7.2)
(The probability that an error within this range will occur is 0.50.)
The probability that an error will lie between two values is given by the
ratio of the area of the probability curve included between the values to the total
area. Inasmuch as the area under the entire probability curve is unity, there is a
100 percent probability that all measurements will lie within the range of the
curve.
The area of the curve between
s
is 0.683; that is, there is a 68.3 percent
probability of an error between 
s
in a single measurement. This error range
is also called the one-sigma or 68.3 percent confidence level. The area of the
curve between  2
s
is 0.955. Thus, there is a 95.5 percent probability of an
error between  2
s
and  2
s
that represents the 95.5 percent error (two-
sigma or 95.5 percent confidence level). Similarly,  3
s
is referred to as the
99.7 percent error (three-sigma or 99.7 percent confidence level). For practical
purposes, a maximum tolerable level often is assumed to be the 99.9 percent
error. Table 7.1 indicates the probability of occurrence of larger errors in a sin-
gle measurement.
The probable error of the combined effects of accidental errors from differ-
ent causes is
(7.3) E
sum

2
E
2
1
 E
2
2
 E
2
3
 
PE
s
0.6745
s

s

B

d
2
n  1
2
1

2
1

6400
1

100
1

400
where E
1
, E
2
, E
3
. . . are probable errors of the separate measurements.
Error of the mean is 
(7.4)
where E
s
 specified error of a single measurement.
Probable error of the mean is 
(7.5)
MEASUREMENT OF DISTANCE WITH TAPES
Reasonable precisions for different methods of measuring distances are
Pacing (ordinary terrain): to 
Taping (ordinary steel tape): to (Results can be improved by use of
tension apparatus, transit alignment, leveling.)
Baseline (invar tape): to 
Stadia: to (with special procedures)
Subtense bar: to (for short distances, with a 1-s theodolite, averag-
ing angles taken at both ends)
Electronic distance measurement (EDM) devices have been in use since the
middle of the twentieth century and have now largely replaced steel tape mea-
surements on large projects. The continued development, and the resulting drop
in prices, are making their use widespread. A knowledge of steel-taping errors
and corrections remains important, however, because use of earlier survey data
requires a knowledge of how the measurements were made, common sources for
errors, and corrections that were typically required.
1

7000
1

1000
1

500
1

300
1

1,000,000
1

50,000
1

10,000
1

1000
1

100
1

50
PE
m

PE
s
n
0.6745
B

d
2
n(n  1)
E
m

E
sum
n

E
s
n
n

E
s
n
TABLE 7.1 Probability of Error in a Single Measurement
Probability
Confidence of larger
Error level, % error 
Probable (0.6745
s
) 50 1 in 2
Standard deviation (
s
) 68.3 1 in 3
90% (1.6449
s
) 90 1 in 10
2
s
or 95.5% 95.5 1 in 20
3
s
or 97.7% 99.7 1 in 370
Maximum (3.29
s
) 99.9 1 in 1000
For ordinary taping, a tape accurate to 0.01 ft (0.00305 m) should be used. The
tension of the tape should be about 15 lb (66.7 N). The temperature should be
determined within 10°F (5.56°C); and the slope of the ground, within 2 percent;
and the proper corrections, applied. The correction to be applied for tempera-
ture when using a steel tape is 
(7.6)
The correction to be made to measurements on a slope is
(7.7)
or (7.8)
or (7.9)
where C
t
 temperature correction to measured length, ft (m)
C
h
 correction to be subtracted from slope distance, ft (m)
s  measured length, ft (m)
T  temperature at which measurements are made, F (C)
T
0
 temperature at which tape is standardized, F (C)
h  difference in elevation at ends of measured length, ft (m)
 slope angle, degree
In more accurate taping, using a tape standardized when fully supported through-
out, corrections should also be made for tension and for support conditions. The cor-
rection for tension is
(7.10)
The correction for sag when not fully supported is 
(7.11)
where C
p
 tension correction to measured length, ft (m)
C
s
 sag correction to measured length for each section of unsupported
tape, ft (m)
P
m
 actual tension, lb (N)
P
s
 tension at which tape is standardized, lb (N) (usually 10 lb) (44.4 N)
S  cross-sectional area of tape, in
2
(mm
2
)
E  modulus of elasticity of tape, lb/in
2
(MPa) [29 million lb/in
2
(MPa) for
steel] (199,955 MPa)
w  weight of tape, lb/ft (kg/m)
L  unsupported length, ft (m)
Slope Corrections
In slope measurements, the horizontal distance H  L cos x, where L 
slope distance and x  vertical angle, measured from the horizontal—a simple
C
s

w
2
L
3
24P
2
m
C
p

(P
m
 P
s
)s
SE
 h
2
/2s  approximate
 0.00015s
2 
approximate
C
h
 s (1  cos )  exact
C
t
 0.0000065s(T  T
0
)
Page 5


SURVEYING
FORMULAS
UNITS OF MEASUREMENT
Units of measurement used in past and present surveys are
For construction work: feet, inches, fractions of inches (m, mm)
For most surveys: feet, tenths, hundredths, thousandths (m, mm)
For National Geodetic Survey (NGS) control surveys: meters, 0.1, 0.01, 0.001 m
The most-used equivalents are
1 meter  39.37 in (exactly)  3.2808 ft
1 rod  1 pole  1 perch  ft (5.029 m)
1 engineer’s chain  100 ft  100 links (30.48 m)
1 Gunter’s chain  66 ft (20.11 m)  100 Gunter’s links (lk)  4 rods  mi
(0.020 km)
1 acre  100,000 sq (Gunter’s) links  43,560 ft
2
 160 rods
2
 10 sq
(Gunter’s) chains  4046.87 m
2
 0.4047 ha
1 rood  acre (1011.5 m
2
)  40 rods
2
(also local unit  to 8 yd) 
(5.029 to 7.315 m)
1 ha  10,000 m
2
 107,639.10 ft
2
 2.471 acres
1 arpent  about 0.85 acre, or length of side of 1 square arpent (varies) (about
3439.1 m
2
)
1 statute mi  5280 ft  1609.35 m
1 mi
2
 640 acres (258.94 ha)
1 nautical mi (U.S.)  6080.27 ft  1853.248 m
1 fathom  6 ft (1.829 m)
1 cubit  18 in (0.457 m)
1 vara  33 in (0.838 m) (Calif.), in (0.851 m) (Texas), varies
1 degree  circle  60 min  3600 s  0.01745 rad
sin 1 0.01745241
1

360
33
1

3
5
1

2
3

4
1

80
16
1

2
1 rad  57 17

44.8

or about 57.30
1 grad (grade)  circle  quadrant  100 centesimal min  10
4
cen-
tesimals (French)
1 mil  circle  0.05625
1 military pace (milpace)  ft (0.762 m)
THEORY OF ERRORS
When a number of surveying measurements of the same quantity have been
made, they must be analyzed on the basis of probability and the theory of
errors. After all systematic (cumulative) errors and mistakes have been elimi-
nated, random (compensating) errors are investigated to determine the most
probable value (mean) and other critical values. Formulas determined from
statistical theory and the normal, or Gaussian, bell-shaped probability distrib-
ution curve, for the most common of these values follow.
Standard deviation of a series of observations is
(7.1)
where d  residual (difference from mean) of single observation and n  num-
ber of observations.
The probable error of a single observation is 
(7.2)
(The probability that an error within this range will occur is 0.50.)
The probability that an error will lie between two values is given by the
ratio of the area of the probability curve included between the values to the total
area. Inasmuch as the area under the entire probability curve is unity, there is a
100 percent probability that all measurements will lie within the range of the
curve.
The area of the curve between
s
is 0.683; that is, there is a 68.3 percent
probability of an error between 
s
in a single measurement. This error range
is also called the one-sigma or 68.3 percent confidence level. The area of the
curve between  2
s
is 0.955. Thus, there is a 95.5 percent probability of an
error between  2
s
and  2
s
that represents the 95.5 percent error (two-
sigma or 95.5 percent confidence level). Similarly,  3
s
is referred to as the
99.7 percent error (three-sigma or 99.7 percent confidence level). For practical
purposes, a maximum tolerable level often is assumed to be the 99.9 percent
error. Table 7.1 indicates the probability of occurrence of larger errors in a sin-
gle measurement.
The probable error of the combined effects of accidental errors from differ-
ent causes is
(7.3) E
sum

2
E
2
1
 E
2
2
 E
2
3
 
PE
s
0.6745
s

s

B

d
2
n  1
2
1

2
1

6400
1

100
1

400
where E
1
, E
2
, E
3
. . . are probable errors of the separate measurements.
Error of the mean is 
(7.4)
where E
s
 specified error of a single measurement.
Probable error of the mean is 
(7.5)
MEASUREMENT OF DISTANCE WITH TAPES
Reasonable precisions for different methods of measuring distances are
Pacing (ordinary terrain): to 
Taping (ordinary steel tape): to (Results can be improved by use of
tension apparatus, transit alignment, leveling.)
Baseline (invar tape): to 
Stadia: to (with special procedures)
Subtense bar: to (for short distances, with a 1-s theodolite, averag-
ing angles taken at both ends)
Electronic distance measurement (EDM) devices have been in use since the
middle of the twentieth century and have now largely replaced steel tape mea-
surements on large projects. The continued development, and the resulting drop
in prices, are making their use widespread. A knowledge of steel-taping errors
and corrections remains important, however, because use of earlier survey data
requires a knowledge of how the measurements were made, common sources for
errors, and corrections that were typically required.
1

7000
1

1000
1

500
1

300
1

1,000,000
1

50,000
1

10,000
1

1000
1

100
1

50
PE
m

PE
s
n
0.6745
B

d
2
n(n  1)
E
m

E
sum
n

E
s
n
n

E
s
n
TABLE 7.1 Probability of Error in a Single Measurement
Probability
Confidence of larger
Error level, % error 
Probable (0.6745
s
) 50 1 in 2
Standard deviation (
s
) 68.3 1 in 3
90% (1.6449
s
) 90 1 in 10
2
s
or 95.5% 95.5 1 in 20
3
s
or 97.7% 99.7 1 in 370
Maximum (3.29
s
) 99.9 1 in 1000
For ordinary taping, a tape accurate to 0.01 ft (0.00305 m) should be used. The
tension of the tape should be about 15 lb (66.7 N). The temperature should be
determined within 10°F (5.56°C); and the slope of the ground, within 2 percent;
and the proper corrections, applied. The correction to be applied for tempera-
ture when using a steel tape is 
(7.6)
The correction to be made to measurements on a slope is
(7.7)
or (7.8)
or (7.9)
where C
t
 temperature correction to measured length, ft (m)
C
h
 correction to be subtracted from slope distance, ft (m)
s  measured length, ft (m)
T  temperature at which measurements are made, F (C)
T
0
 temperature at which tape is standardized, F (C)
h  difference in elevation at ends of measured length, ft (m)
 slope angle, degree
In more accurate taping, using a tape standardized when fully supported through-
out, corrections should also be made for tension and for support conditions. The cor-
rection for tension is
(7.10)
The correction for sag when not fully supported is 
(7.11)
where C
p
 tension correction to measured length, ft (m)
C
s
 sag correction to measured length for each section of unsupported
tape, ft (m)
P
m
 actual tension, lb (N)
P
s
 tension at which tape is standardized, lb (N) (usually 10 lb) (44.4 N)
S  cross-sectional area of tape, in
2
(mm
2
)
E  modulus of elasticity of tape, lb/in
2
(MPa) [29 million lb/in
2
(MPa) for
steel] (199,955 MPa)
w  weight of tape, lb/ft (kg/m)
L  unsupported length, ft (m)
Slope Corrections
In slope measurements, the horizontal distance H  L cos x, where L 
slope distance and x  vertical angle, measured from the horizontal—a simple
C
s

w
2
L
3
24P
2
m
C
p

(P
m
 P
s
)s
SE
 h
2
/2s  approximate
 0.00015s
2 
approximate
C
h
 s (1  cos )  exact
C
t
 0.0000065s(T  T
0
)
hand calculator operation. For slopes of 10 percent or less, the correction to be
applied to L for a difference d in elevation between tape ends, or for a horizon-
tal offset d between tape ends, may be computed from
(7.12)
For a slope greater than 10 percent, C
s
may be determined from
(7.13)
Temperature Corrections
For incorrect tape length:
(7.14)
For nonstandard tension:
(7.15)
where A  cross-sectional area of tape, in
2
(mm
2
); and E  modulus of elas-
ticity  29,000,00 lb/in
2
for steel (199,955 MPa).
For sag correction between points of support, ft (m):
(7.16)
where w  weight of tape per foot, lb (N)
L
s
 unsupported length of tape, ft (m)
P  pull on tape, lb (N)
Orthometric Correction
This is a correction applied to preliminary elevations due to flattening of the
earth in the polar direction. Its value is a function of the latitude and elevation
of the level circuit.
Curvature of the earth causes a horizontal line to depart from a level sur-
face. The departure C
f
, ft; or C
m
, (m), may be computed from
(7.17)
(7.18) C
m
 0.0785K
2
C
f
 0.667M
2
 0.0239F
2
C 
w
2
L
3
s
24P
2
C
t

(applied pull  standard tension)L
AE
C
t

(actual tape length  nominal tape length)L
nominal tape length
C
s

d
2
2L

d
4
8L
3
C
s

d
2
2L