Lesson 2: DISTANCE MEASUREMENT (Horizontal)The distance between any two random points in three dimensional space is a spatial distance. There are several methods of determining distance, the choice of which depends on the accuracy required, the cost, and other conditions. The methods in ascending order of accuracy are estimation, scaling from a map, pacing, odometer, tacheometry, taping, photogrammetry, and electronic distance measurement.METHODS OF MEASUREMENTPacingPacing consists of counting the number of steps, or paces, in a required distance. The length of an individuals pace must be determined first. Pacing furnishes a rapid means of approximately checking more precise measurements of distance. It is used on reconnaissance surveys and, in small-scale mapping, for locating details and traversing with the plane table. Pacing over rough country has furnished a relative precision of 1/100; under average conditions, a person of experience will have a little difficulty in pacing with a relative precision of 1/200.

Pace Distance = Pace Factor x No. of Paces

Pace Factor = Known Length of Distance ABMean Number of Paces for AB

Mileage recorder, odometer, and other methods Distance maybe measured by observing the number of revolutions of the wheel of vehicle. The mileage recorder attached to the ordinary automobile speedometer registers distance to 0.1 mi and may be read by estimation to 0.01 mi. Special speedometers are available reading to 0.01 or 0.002 mi. The odometer is a simple device that can be attached to any vehicle and directly registers the number of revolutions of a wheel. With the circumference of the wheel known, the relation between revolutions and distance is fixed. The distance indicated by either the mileage recorder or the odometer is somewhat greater than the true horizontal distance, but in hilly country a rough correction based on the estimated average slope may be applied.TacheometryTacheometry includes stadia with transit and stadia rod; stadia with alidade, plane table, and rod; distance wedge and horizontal rod; and subtense bar and theodolite.TapingTaping involves direct measurement of the distance with steel tapes varying in length from 3ft (1 m) to 1000 ft (300 m). Graduations are in feet, tenths, and hundredths, or metres, decimeters, centimeters, and millimeters. The precision of distance measured with tapes depends upon the degree of refinement with which measurements are taken. On the one hand, rough taping through broken country may be less accurate than the stadia. On the other hand, when extreme care is taken to eliminate all possible errors, measurements have been taken with a relative precision of less than 1/1,000,000. In ordinary taping over flat, smooth ground, the relative precision is about 1/3000 to 1/5000.Electronic distance measurement Recent scientific advances have led to the development of electro-optical and electromagnetic instruments which are of great value to the surveyor for accurate measurements of distances. Measurement of distance with electronic distance measuring (EDM) equipment is based on the invariant speed of light or electromagnetic waves in a vacuum. EDM equipment which can be used for traverse, triangulation, and trilateration as well as for construction layout is rapidly supplanting taping for modern surveying operations except for short distances and certain types of construction layout.

Choice of methods Most boundary, control, and construction surveys involving long lines and large areas can be performed most accurately and economically using modern EDM equipment. Where the distances involved are relatively short or specific construction layout requirements are present, taping the distances can be more practical. Stadia is still unsurpassed for small topographic surveys and preliminary surveys for projects of limited extent.Each of the methods mentioned in the preceding sections has a field of usefulness. On the survey for a single enterprise, the surveyor may find occasion to employ a combination of methods to advantage.TapesTapes are made in a variety of materials, lengths, and weights. Those more commonly used by the surveyor and for engineering measurements are the steel tapes, sometimes called the engineers or surveyors tape, and woven nonmetallic and metallic tapes.Errors in measurement of Distances1. Tape not standard length2. Imperfect alignment of tape3. Tape not horizontal4. Tape not stretch straight5. Imperfection of observation6. Variations in temperature7. Variations in tensionMistakes in Measurement of Distances1. Adding or dropping a full tape length.2. Adding a cm., usually in measuring the fractional part of tape length at the end of the line.3. Recording numbers incorrectly, example 78 is read as 87.4. Reading wrong meter mark.Correction applied for measurement of distances1. Temperature correction: (To be added or subtracted)Ct=K (T2-T1) L1

K=0.00000645 ft. per degree FK=0.0000116 m. per degree CT1=temp. when the length of tape isL1T2=temp. during measurement

2. Pull Correction: (to be added or subtracted)Cp= (P2-P1) L1AEP2=actual pull during measurementP1=applied pull when the length of tape is L1A=Cross-sectional area of tapeE=Modulus of elasticity of tape

3. Sag Correction: (To be subtracted only)Cs=w2L3 24 P2

w=weight of tape in plf. or kg. m.L=unsupported length of tape p=actual pull or tension applied

4. Slope Correction: (To be subtracted only)Cs= h2 2 S

H=S CsH=horizontal distance or corrected distanceS=inclined distanceh=difference in elevation at the end of the tape

5. Sea level correction:Reduction factor = 1 h R

B1 = B (1- h) R

B= horizontal distance corrected for temperature, sag and pull.B1 = sealevel distanceh=average altitude or observationR=Radius of curvature

6. Normal Tension:It is the tension which is applied to a tape supported over two supports which balances the correction due to pull and due to sag. The application of the tensile force increases the length of the tape whereas the sag decreases its length, the normal tension neutralizes both corrections, therefore no correction is necessary.

PN = 0.204 WAE PN P1P= applied normal tensionP1= tension at which the tape is standardizedW= total weight of tapeA= cross-sectional area of typeE= modulus of elasticity of tape

Sample Problem No. 1:A line 100-m long was paced by a surveyor for four times with the following data: 142, 145, 145.5 and 146. Then a new line was paced for four times again with the following results, 893, 893.5, 891 and 895.5.a. Determine the pace factor.b. Determine the distance of the new line.

Sample Problem 2:A 50 m tape was standardized and was found to be 0.0042 m too long than the standard length at an observed temperature of 58C and a pull of 15 kilos. The same tape was used to measure a certain distance and was recorded to be 637.92 m long at an observed temp. of 68 C and a pull of 15 kilos. Coefficient of thermal expansion is 0.0000116 m/ C.a. Determine the standard temperature.b. Determine the total correction.c. Determine the true length of the line.

Sample Problem 3:A line is recorded as 472.90 m long. It is measured with a 0.65 kg tape which is 30 m long at 20C under a 50 N pull supported at both ends. During measurement, the temperature is 5 C and the tape is suspended under a 75 N pull. The line is measured on 3% grade. E = 200 Gpa, cross-sectional area of tape is 3 mm2 and the coefficient of thermal expansion is 0.0000116 m/ C. a. Compute the actual length of tape during measurement.b. Compute the total error to be corrected for the inclined distance.c. What is the true horizontal distance?

OTHER SURVEYS WITH TAPEThe tape is not necessarily limited only to the measurement of distances. There are various problems arising in surveying fieldwork which can be solved just by the use of a tape. Some of these surveying operations include: erecting perpendicular to a line, measuring angles, laying off angles, determining obstructed distances, locating irregular boundaries, and determining areas of different shapes.1. Erecting Perpendicular To a LineThere are some instances when it would be necessary to erect on the ground a perpendicular to an established line. For example, when the floor dimensions of a building or a road intersection are to be laid out, it becomes necessary to erect perpendicular lines. Commonly employed for such particular requirements are the chord-bisection method and the 3:4:5 method.a. Chord-Bisection MethodIn the figure shown, it is required to erect a perpendicular to the line AB at point m. Two equal lengths, mb and mc, are measured on each side of point m. With b as center and taking any convenient length of tape as radius, an arc of a circle is described. The same procedure is repeated at point c. The intersection of the two arcs locates point d, and line dm is the desired perpendicular to AB.

b. 3:4:5 MethodThis method of erecting a perpendicular line is illustrated in the figure. The method involves the setting up on the ground of a triangle whose three sides are made in the proportions of 3, 4, and 5. Point A is selected on line MN where a perpendicular is to be erected. From A and along line MN, measure 3.0 m to the first tapeman at B and the 10-m mark held by the second tapeman at A, a loop is formed by the third tapeman to bring the 5-m and 6-m marks together. The third tapeman then pulls each part of the tape taut to locate point C on the ground. The line joining points A and C is the desired perpendicular to line MN.Any other lengths in the proportions of 3, 4, and 5 can also be used such as 6:8:10, 9:12:15, and 12:16:20.

Chord Bisection Method3:4:5 Method

2. Measuring Angles with TapeA tape is not frequently used in engineering constructions for measuring or laying out angles. There may be some occasions, however, when a theodolite or a transit is not readily available that the tape is used instead. The measurement of very small angles with tape usually gives satisfactory results. All angular measurements by tape are accomplished by the application of very basic geometric and trigonometric principles.a. In the figure shown, let it be required to measure the angle BOC (or ). One way to do it is to lay out any convenient length (L) along lines OB and OC to establish points a and b. If the chord distance ab (or d) is measured and bisected, the angle BOC can be computed as follows. Sin(/2) = d/2 L

b. If unequal lengths are laid out, as AB and AC in the figure shown, then upon measuring the distance BC, the angle BAC (or < A) can be computed using law of cosine.

Cos A = (AC)2 + (AB)2 - (BC)22(AC)(AB)

3. Determining Obstructed DistancesIn some instances is may not be possible to directly measure distances due to an obstruction. The required length may also be inaccessible or difficult to measure. The following are some of the indirect methods which could be employed to determine obstructed or inaccessible distances.

a. By Right Angle

In figure shown, point C is established at a convenient location away from the obstruction and it is seen to it that lines AC and BC intersect at right angles. Both lines are measured accurately as possible. If A and B define the end points of the required line, the length of line AB can be computed by the Pythagorean Theorem:AB = (AC2 + BC2)

b. By Swing Offsets

The line AB in the figure shown could not be determined because of an obstruction somewhere at the middle of the line. To determine its length, the head tapeman anchors one end of the tape at B and swings it using any convenient radius. The rear tapeman positions himself at point A and lines in the other end of the tape with a distant point as D and directs the marking of points a and b on the ground where the end of the tape crosses line AD. The midpoint of line ab is located to establish point C.With line BC established perpendicular to line AC, the length of AB can be indirectly determined also by the Pythagorean Theorem since AC and BC are known.

c. By Parallel Lines

If the necessary distance from line AB is short, perpendicular line AA = BB are erected by either using the chord-bisection method or the 3:4:5 method to clear the obstacle. The line AB is then taped, and its length is taken as that of AB.

d. By Similar TrianglesThe method illustrated in the figure is one where two line, BD and CE, are established perpendicular to the line ABC. The distance between the two perpendiculars (or BC) is measured and with points D and E both line up with A, the length of AB can then be determined by similarity of triangles or:

AB = (AB + BC) ; AB (CE) = BD (AB + BC)BD CE

AB (CE) = BD (AB) + BD(BC)AB (CE) - BD (AB) = BD(BC)

AB (CE - BD) = BD (BC) orAB = BD (BC) (CE - BD)

SAMPLE PROBLEMS:1. The angle between two intersecting fences is to be determined with a tape. A point on each fence line is established 30.0 m from the point of the intersection. If the distance between the established points is 12.20 m, what is the intersection angle?

2. In the quadrilateral ABCD shown, the following lengths were measured by tape: AB = 760.50 m, BC = 390.80 m, CD = 371.60 m, DA = 598.80 m and AC = 765.40 m. Compute the interior angle at each corner.

3. In the figure shown, lines NQ and PR are established perpendicular to line MNP, and points Q and R are lined up with the distant point M. If NQ = 318.55 m, PR = 475.62 m, and NP = 210.38 m, determine the length of MN which represents the width of the river.