Theory of Errors and AdjustmentsLecture 2

Caraga State UniversityCollege of Engineering and Information Technology

Broddett B. Abatayo, GE Lecturer

Rules for Applying Tape Corrections

Tape too long:1. Add correction when measuring distances2. Subtract correction when laying out distances

Tape too short:1. Subtract correction when measuring distances2. Add correction when laying out distances

Measure Laying out

Too long + -

Too short - +

A B

MEASURE

Standard

Too short

Too long

Distance AB = 8cm

8cm

LAYING OUT

Standard

Too short

Too long

Taping CorrectionsA. Correction due to Temperature: (to be added or subtracted)

Where;α = coefficient of thermal expansion (0.0000116/°C)To = observed temperature during measurementTs = standard temperatureL = Nominal length of tape or total measured distance

Example:

When the temperature was 48°C, the measured distance from B to C was 318 m. The steel tape used has a standard length at 20°Cm with a coefficient of thermal expansion of 0.0000116/°C. Find the correct distance BC in meters.

Ans. 318.103 m

LTsToC t )(

B. Correction due to Pull: (To be added or subtracted)

Where;Po = applied pull during measurementPs = standard pullL = Nominal length of tape or total

measured distanceA = cross-sectional area of tapeE = modulus of elasticity of tape

Taping Corrections

AE

LPsPoCp

)(

Example:

A 50 m tape having a cross-sectional area of 0.05cm2 has been standardized at a tension of 5.5kg. If the modulus of elasticity E = 2.10x106 kg/cm2, determine the elongation of the tape if a pull of 12 kg. is applied.

Ans. 0.003m

5.5 kg5.5 kg

12 kg 12 kg

standardition

50m

During measurement

Taping CorrectionsC. Correction Due to Sag: (to be subtracted only)

Where: ω = weight of tape per unit length W = total mass or weight of tape L = unsupported length of tape Po = applied pull during measurement

2

2

2

32

2424 Po

LW

Po

LCsag

Example

A 30m tape is supported only at its ends and under a steady pull of 8kg. If the tape weighs 0.91kg. Determine the following:

1. Sag correction2. Correct distance between the ends of

the tape

Ans. 0.0162m, 29.984m

Taping CorrectionsD. Combined correction:

E. Correction due to slope: (to be subtracted only)

Where;S = inclined/slope distanceH = correct horizontal distanceh = vertical distance at ends of tape

during measurement

Taping Corrections

S

hCs

2

2

CsSH

Example

Slope distances AB and BC measure 330.49m and 660.97m, respectively. The difference in elevation is 12.22m for A and B, 10.85m for B and C. Using the slope correction formula, determine the horizontal length of line ABC. Assume the line AB has a rising slope and BC a falling slope.

Ans. 991.145m

CAB

Taping CorrectionsF. Normal Pull or Tension – The required

pull/tension to eliminate the effect of sag.

CsagCp

2

2

24

)(

Pn

LW

AE

LPsPn

)(204.0

PsPn

AEWPn

Taping CorrectionsG. Reduction to Sea-Level

Where;D = measured distance bet. Two pointsD’ = corresponding sea-level dist. Of

these pointsR = average radius of curvature(1-h/r) = sea-level reduction factorh = observed height

R

hDD 1'

BA

R

Prob 1• When the temperature was 48°C, the

measured distance from B to C was 318 m. The steel tape used has a standard length at 20°Cm with a coefficient of thermal expansion of 0.0000116/°C. Find the correct distance BC in meters.

Ans. 318.103 m

Prob 2• When the temperature was 3°, the

distance from E to F was measured using a steel tape that has a standard length at 20 °C with a coefficient of thermal expansion of 0.0000116/ °C. If the correct distance from E to F is 836.5m, what was the measured distance in meters?

Ans. 836.665 m

Prob 3• A 50 m tape was standardized and

was found to be 0.0042m too long than the standard length at an observed temperature of 58 °C and a pull of 15kg. If the same tape was used to measure a certain distance and was recorded to be 673.92m long at an observed temperature of 68 °C and a pull of 15kg, and the coefficient of thermal expansion is 0.0000116/ °C, determine the following:

1. Standard Temperature2. Total correction3. True length of the line

Ans. 50.76 °C, 0.1348m, 674.05 m

Prob 4• A 50 m tape having a cross-sectional

area of 0.05cm2 has been standardized at a tension of 5.5kg. If the modulus of elasticity E = 2.10x106 kg/cm2, determine the elongation of the tape if a pull of 12 kg. is applied.

Ans. 0.003m

Prob 5• It takes 20 kg of normal tension to

make the elongation of a steel tape offset the effect of sag when supported at the end points. The tape has a cross-sectional area of 0.05cm2 and E = 2x106 kg/cm2. If the tape is 50m long and has a standard pull of 8kg. What is its unit weight in kg/m?

Ans. 0.0215 kg/m

Prob 6• A 30m tape is supported only at its

ends and under a steady pull of 8kg. If the tape weighs 0.91kg. Determine the following:

1. Sag correction2. Correct distance between the ends of

the tape

Ans. 0.0162m, 29.984m

Prob 7• A 100m tape weighs 0.0508 kg/m.

During field measurements, the tape was subjected to a tension of 45 N, and was supported at the end points, midpoint, and quarter points, find the correction per tape length due to sag.

Ans. 0.319 m

Prob 8• Slope distances AB and BC measure

330.49m and 660.97m, respectively. The difference in elevation is 12.22m for A and B, 10.85m for B and C. Using the slope correction formula, determine the horizontal length of line ABC. Assume the line AB has a rising slope and BC a falling slope.

Ans. 991.145m

Prob 9• A line 100 m long was measured with

a 50m tape. It was discovered that the first pin was stuck 30cm to the left of the line and the second pin 30cm to the right. Find the error in the measurement in cm.

Ans. 0.45cm

Prob 10• A line was determined to be

2395.25m when measured with a 30m steel tape supported throughout its length under a pull of 4kg at a mean temperature of 35°C. The tape used is of standard length at 20°C under a pull of 5kg. If the cross-sectional area of the tape is 0.03cm2, coefficient of thermal expansion is 0.0000116/°C, and E = 2x106 kg/cm2, determine the following:

1. Temperature correction2. Pull correction3. Correct length of the line

Ans. +0.4168m, -0.0399m, 2395.6269m