LAB EXERCISE NO. 2 PROBLEM
: TAPING OVER SMOOTH AND LEVEL GROUND
OBJECTIVES : a) To determine the horizontal length of the line over smooth and level ground with the tape supports throughout its length. LAB EXERCISE OUTLINE A. INSTRUMENTS & ACCESSORIES : 30-m or 100-ft Steel tape, Chaining Pins, Range pole, Pegs or Hubs Chalk or Marking Crayons, and Spring Scale. B. PROCEDURE :
1. Using hubs, mark both ends of the line (150 to 300 m long) to be measured. Place a range pole behind each point and also establish intermediate points along the line to ensure unobstructed sight lines. If the point to be marked is on a pavement, pavement, use use chalk or marking crayons. crayons. Designate Designate these these end points as A and B. 2. The rear tape man with one pin stations himself at the point of beginning and the head tape man takes the zero end of the tape and advance toward the other end of the line to be measured. 3. When the head tape man has gone nearly a full tape length, the rear tape man calls “tape” to stop the head tape man. The rear tape man then holds the 30-m (or 100 ft) mark at the starting point and aligns the other end of the tape held by the head tape man on the range pole set behind the end point or on any of the intermediate points earlier established along the line. 4. With the 30-m mark at a t the starting point, and the head tape man aligned, the rear tape man calls “all right”. The head tape man then pulls the tape taut and sticks a chaining pin in the ground to mark the 0 -m end of the tape. If the measurement is done on pavement, a chalk or marking crayon is used to mark the end of the tape. 5. The rear tape man picks up his chaining pin and the head tape man pulls the tape forward and the process is repeated for the next full tape length. 6. When the end of the line is almost reached and the last full tape length has been measured, the remaining partial length length is then then measured. measured. The rear tape man holds holds the tape until he has a full meter (or foot) foot) mark at at his chaining pin while the head tape man pulls the tape taut and takes note of the fractional measurement read form the tape end. 7. After the measurement of the whole line is completed, a second measurement should be made along the opposite direction. The mean of the two measurements is taken as the most probable value of the length of the line. 8. Tabulate observed and computed values accordingly. Refer to the accompanying sample format for tabulation of data. Trail 1 2
Line AB BA
Length
Difference
Mean
Relative Precision
C. COMPUTATIONS :
1. The mean length of t measured line is determined by adding the two measurements and dividing the sum two. 2. The discrepancy for the measurement is the difference between the first measurement and the second measurement. 3. The relative precision of the measurement is determined by dividing the discrepancy by the mean length, where the numerator is reduced to unity or 1. Relative precision is expressed in fractional form and the denominator is rounded to the nearest hundredth.
D. REMARKS, HINTS & PRECAUTIONS :
1. The hub is set at each end of the designated line, and a range pole is set about meter beyond the far hub to serve as guide during taping. A 2”x2” piece of lumber with nail in the middle is used for tis propose and embedded securely into the ground. 2.
Prior to measurement the tape should first be calibrated.
3. As a rough check, the designated line should also be measured by pacing. 4. During measurement the tape should be supported throughout its entire length and a required pull (approximately 4 to 7 kg) applied at the end of tape to avoid inconsistent errors. 5. During the line-in process, the head tape man is in a kneeling position to one side and facing the line so that he can hold the tape steady and allow the rear tape man to have a clear view of the range pole marking the distant point. 6. As the rear tape man leaves after each full tape length is measured, a pin is pulled out of the ground. There is always one pin in the ground. There is always one pin in the ground, and the number of pins held by the rear tape man at any time indicates the number of full tape lengths measured from the point of beginning to the pin in the ground. 7. A steady and firm pull is made on one end of the tape if the leather thong at the end of the tape is wrapped around the hand, or by holding on to a changing pin slipped through the eye at the end of the tape, or by using a tape clamp. 8. The take me be held between the fleshy portion of the fingers and that of the palm. Enough friction is applied to sustain the required pull without causing injury or discomfort to the tape man. 9. The acceptable precision should be at least 1/1000. If this is not attending the measurement should be repeated.
LAB EXERCISE NO. 3 PROBLEM: DETERMINING AREA OF A RECTILINEAR FIELD BY TAPE OBJECTIVES:
a) To learn how to measure horizontal angles with tape. b) To determine the area of a rectilinear field with tape. ------------------------------------------------------------------------------------------------------------------------------LAB EXERCISE OUTLINE A. INSTRUMENTS & ACCESSORIES: Steel Tape, Chaining Pins, Range Poles, and Hubs or Pegs B. PROCEDURE: 1. Establish the corners of the assigned field and also establish within the field a centrally-located point. 2. Subdivide the field into a convenient series of connected triangles (refer to accompanying figure). Use pegs, hubs, or pins to mark the vertices to each triangle. Call these points A, B, C, and etc. 3. From centrally-located point, measure distances to each po int defining the corners or vertices of each triangle. Call these distances d1, d2, and etc. 4. Using the chord method of measuring angles by tape, determine all angles about the central point, i.e., Ө1, Ө2, and etc. Fig. 4-3. A field subdivided into triangles.
5. Refer to the accompanying sample tabulation for the recording of observed field data. C. COMPUTATIONS: 1. Determining Value of Included Angles
With the use of a tape the chord method of measuring angles may be applied to determine the value of an angle. The following formula is used.
Sin Ө/2 =
d/2 L
Where: Ө = angle whose value is desired d = measured chord distance L = any convenient length of tape swung through an arc
Fig. 4-4. Chord method of measuring angles
INCLUDED ANGLE Ө1 Ө2 Ө3 Ө4 Ө5
CHORD DIST
LENGTH OF SIDE
COMPUTED ANGLE
CORR
ADJUSTED ANGLE
2. Determining Area of Each Triangle In each triangle, since the length of two sides and the included angle can be determined, the area of the triangle may be calculated by the following formula A = (1/2)(a)(b)Sin C Where: A = area of the triangle a, b = the two measured sides of the triangle (or d 1, d2, etc.) C = included angle (or Ө 1, Ө2, Ө3, etc.)
TRIANGLE
SIDES 1st
2nd
INCLUDED ANGLE
AREA
ABF BCF CDF DEF AEF 3. Calculating the Total Area The total area of the field is the sum of the calculated areas of the series of connected triangles into which the field is sub-divided, or AT = A1 + A2 + A3 + A4 + A5
WHERE: AT = total area of the field, and A 1, A2, and etc are areas of individual triangles.
D. REMARKS, HINTS & PRECAUTIONS: 1. Determining areas of rectilinear fields by tape is a slow process and is not advisable when large areas are involved. 2. This method of field survey was widely used b efore instruments for measuring angles were built. 3. The accuracy of the values obtained by this procedure is dependent on the size of the angles, on the care with which the points are set on line, and on the accuracy of the measured lengths. 4. Experience will show that with reasonable care the value of an angle determined by tape measurements would agree with the value obtained with a transit within one to three minutes. 5. All distances from the centrally-located point of the field to the different corners should be measured twice and the mean recorded as its actual length.
LAB EXERCISE NO. 4 PROBLEM : DIFFERENT LEVELING OBJECTIVE: To determine the elevation of points by differential leveling. A. INSTRUMENTS &ACCESSORIES : Dumpy or Wye level, Leveling Rod, Hubs or Pegs, Range Poles, Hammer or Mallet, and Chalk. B. PROCEDURE :
1. Designate the initial or reference point of the level route as BM-1 and the final or terminal point as BM2. 2. Set up and level the instrument at a convenient location along the general designated route between two points. 3. Take and record a backsight on BM-1. 4. Direct the rodman to move forward along the level route (not necessarily in a straight line between BM1 and the next point) and establish TP-1. 5. Take and record foresight in TP-1. 6. Transfer and set up the level at another suitable location beyond TP-1 and take and record a backsight on TP-1. 7. Direct the rodman to move a convenient distance ahead of the instrument and establish TP-2. 8. Take and record a foresight on TP-2. 9. Repeat the procedure until a foresight is finally taken on BM-2. 10. Refer to the accompanying sample tabulation for the recording of observed field data. STA
BS
HI
FS
ELEVATION
C. COMPUTATIONS:
1. Computing Height of Instrument and Elevation. In differential leveling, the following two equations are repeatedly used a) HI = Elev + BS
b) Elev = HI - FS
Where: HI = height of instrument above a reference datum BS = backsight reading on the rod (or plus sights) FS = foresight reading on the rod (or minus sights) Elev = elevation above a reference datum of a point sighted
2.
Arithmetic Check.
REMARKS
To check the accuracy of the arithmetical computations, add all backsights as well as all the foresights. The difference between these two sums must be equal to the difference in elevation between the initial and final points of the level line. Fig. 4-5 Differential Leveling
D. REMARKS, HINTS & PRECAUTIONS :
1. To reduce to a neglible amount the effect of curvature and refraction, keep the backsight distance approximately equal to its corresponding foresight distance. 2. Limit foresight and backsight distances to a maximum of 90 meter since most leveling rods cannot be read accurately beyond this distance. 3. Always keep the telescope bubble centered at instant of sighting. The accuracy of leveling depends on the bubble being in this position during each rod sighting. 4. Avoid reading too near the top or bottom of the rod. Select suitable locations for the instrument so that backsight and foresight rod readings will fall between 0.50 and 1.50 m marks of the leveling rod. 5. Avoid the use of a long (extended) rod especially when it is windy and the leveling rod is not provided with a rod support. 6. Since more time is spent at a set up of the instrument, care must be observed so as to prevent settlement of the instrument between backsighting and foresighting periods. To guard against settlement of the instrument, such as when working on soft grounds during rainy days, the tripod legs must be driven firmly into good bearing soil. 7. Turning points should have a rounded or leveled top to give a definite point of support for the bottom of the leveling rod. If a hub or peg is used, avoid locating such points on soft and unstable grounds. 8. The rod should be held plumb and steady when a reading is made on it. Preferably, use a rod level to assure that the rod will always be held plumb and keep fingers off the face of the rod. 9. Use the proper tension on the leveling screws. They should have bearing on the tripod head without being too tight. 10. For non-precise surveys, rod readings may be made to the nearest hundreth of a meter. 11. The number of instrument set-ups will depend largely on the configuration of the terrain, length of the level circuit or route, and the length of sight distances. It is best to select the shortest level route possible to reduce the amount of error closure.
LAB EXERCISE NO. 5 PROBLEM
: CLOSED COMPASS TRAVERSE
OBJECTIVES : a) To determine the magnetic bearings of each line of a closed compass traverse.
b) To learn how to adjust a closed compass traverse.
LAB EXERCISE OUTLINE A. INSTRUMENTS & ACCESSORIES : Surveyor’s or Brunton Pock et Compass, Pegs or Chaining Pins,
Steel Tape, Range Poles, and Hubs. B. PROCEDURE :
1. Establish at least five traverse stations at designated points on the field assigned to be surveyed. Call these stations (or corners) A, B, C, and so forth. 2. Set and level the compass at A and release the needle of the compass so that it will swing freely on its pivot. 3. Sight the compass on the last traverse station and read the magnetic bearing. Record this as the back bearing of the last line in the traverse. 4. Sight B and read the compass box. Record this as the forward bearing of line AB.
Fig. 4-10. Closed compass traverse.
5. Transfer the compass to B. Sight on A and read the compass box and record this as the back bearing of line AB. 6. Turn towards the direction of C, then read and record the forward of line BC. 7. Transfer to C and sight on B. Read and record the back bearing of line BC, then sight on D and also read and record the forward of bearing of line CD.
8. Proceed to the next station and repeat the above procedure of determining back and forward bearings for each line. Continue until the last traverse station is occupied and the forward bearing of the last line has been read and accordingly recorded. 9. With a steel tape, measure the length of each line in the traverse tice and record the mean as the actual length of the line. 10. Tabulate observed and adjusted values accordingly. Refer to the accompanying sample format for the tabulation of field data.
Note : The following data are indicated in the illustrated sample closed compass traverse : a) The mean length of each traverse line is shown as d1, d2, d3, etc, b) Angular value corresponding to observed bearings are represented by the angle theta (0) All angle with odd-numbered subscripts correspond to forward bearings and angles with even-numbered subscripts correspond to back bearings., c) The interior angle are similarly represented by the angle theta but with subscripts shown in letters of the alphabet such as 0 a, 0 b, 0 c, etc.
OBSERVED BEARINGS LINE
LENGTH
STA FORWARD
AB BC CD DE EA C. COMPUTATION
COMPUTED INT. ANGLE
CORR.
ADJUSTED INT. ANGLE
BACK A B C D E
:
1. The interior angle at each station of the traverse can be computed easily from the observed forward and backward bearings taken from the station regardless of whether or not the needle is affected locally. A carefully prepared sketch is important since from it the required calculations will be obvious. 2. The angular error of closure is determined by subtracting the sum of the computed interior angles of the traverse from (n-2) x 180 deg, where n is the number of sides in the traverse. Note that the computvd value may be a positive or negative quantity, and the resulting sign should be considered in all succeeding computations 3. The correction of each computed interior angle is determined by dividing the angular error of closure by the number of traverse stations. This computed value is then added algebraically to each computed interior angle to determine the corresponding adjusted interior angle. 4. Go over observed bearings and determine which line in the traverse is free from local attraction or which could be arbitrarily chosen as the best line. The adjustment of traverse lines affected by local attraction is then made by starting from the unaffected or best line. 5. The forward and back bearings of all other lines in the traverse affected by local attraction can then be easily computed and adjusted by again drawing a sketch thus making all calculations obvious. 6. Finally, tabulate the adjusted bearings of the traverse. Refer to the accompanying sample format below.
ADJUSTED BEARINGS LINE
LENGTH FORWARD
AB BC CD DE EA
D. REMARKS, HINTS & PRECAUTIONS:
Same as those given in Lab Exercise No. 10.
BACK
LAB EXERCISE NO. 6 PROBLEM
: GRAPHICAL ADJUSTMENT OF CLOSED TRAVERSE
OBJECTIVES : To learn how to plot a closed traverse and adjust the error of closure by a graphical
application of the compass rule.
LAB EXERCISE OUTLINE E. INSTRUMENTS & ACCESSORIES : Protractor, Triangles, Paper, Pen, Pencil, and other Drawing
Instruments F. PROCEDURE :
1. This problem is an indoor lab exercise and it will be assumed that the given data were taken from an actual field observation. 2. The lab instructor has the option to use the prepared data, revise it, or give a few set of data. 3. In order that the students will better understand the procedure involved in working out this lab exercise, the following illustrative problem is given. 4. Illustrative Problem: a) Constructing the Correction Triangle. 1) Using the same scale used in the given traverse, construct a straight line AA’ representing total length of the given traverse and mark off distances AB, BC, CD, DE and EA’ equal to the respective lengths of the different lines in the traverse (Refer to the accompanying sketches). 2) At point A’ of the constructed line draw a perpendicular line A’a equal in length to the error of closure, AA’, as shown in the plotted traverse. 3) From point a on the constructed perpendicular, draw a straight line towards A, then draw lines Bb, Cc, Dd, and Ee parallel to A’a. Each of these lines represents the correction to be applied at points B, C, D, and E respectively, on the plotted traverse. b) Adjusting the Traverse. 1) At point B on the plotted traverse, draw a line parallel to AA’ (error of closure). Draw the line from B in the same direction taken by A’A. 2) Measure Bb on the correction triangle and lay out, starting are station B of the plotted traverse, the same length along the line just drawn through the station. Label this line also as Bb. 3) Repeat the foregoing procedures at traverse stations C, D, and E to similarly lay out the lengths of Cc, Dd, and Ee accordingly.
4-7. Graphical adjustment of a closed traverse.
4-18. The correction triangle. 4) The adjusted traverse is determined by successively connecting the following points now plotted on the traverse: A, b, c, d, e, and back to A. This is shown by dashed lines. 5) After understanding the above illustrated process, consider the following listed two sets of data for closed traverse. It will be expected that an error of closure exists in both data sets. You are now required to:
a) Plot both traverses separately on 8 ½ “ by 11” paper using a scale of 1:100. b) Construct the corresponding correction triangle for each traverse. c) Adjust graphically the error of closure of each traverse. NOTE: Label the plotted traverse and their corresponding correction triangles accordingly. All construction lines should be shown. 6) Data for 1st traverse: The length and bearing of each line are: AB 69.50 m, S73 30’E; BC, 123.00 m, N79 00’E; CD, 80.00 m, N40 00’W; DE, 104.00 m, N86 40’W; and EA, 90.50 m, S33 00’W. 7) Data for the 2nd traverse: The length and azimuth from South of each line are: AB, 83.10 m, 162'30'; Bc, 191.75 m, 265'30'; Cd, 116.50 m, 1'15'; De, 95.00 m, 121'00', and EA, 99.00 m, 76'30'.
C. COMPUTATIONS: Since the solution is graphical, there are no mathematical computations to be made. D. REMARKS, HINTS & PRECAUTIONS: 1. The difference between the initial and the final positions for the first traverse station (A) represents the error of closure for the traverse. This is represented by distance AA' or (e), where A and A' are the initial and final locations of the first station. 2. A suitable scale should be used in plotting the traverse and in drawing the correction triangle. 3. When drawing the correction triangle, if its length could not be contained within the size of paper used, a different scale may be selected, however, the length of A'a (or e) should remain to be drawn equal to the linear error of closure. 4. This graphical adjustment of the error of closure is an application of the compass rule. It will be noted that the distances Aa, Bb, Cc, Dd, and Ee on the correction triangle, are in proportion to the dis from point A to the successive points of the traverse.
LAB EXERCISE #7 OUTLINE PROBLEM: AREA OD A TRACT OF LAND BY DMD/DPD METHOD OBJECTIVES: a) To adjust the latitudes and departures of a closed traverse by the compass rule and determine the area by the double meridian distance method. b) To check the correctness of the area computation by the double parallel distance method.
---------------------------------------------------------------------------------------------------------------------------------------------------------------A. INSTRUMENT & ACCESSORIES: Electronic Calculator, Drawing Instruments, and Graphing or Tracing Paper. B. PROCEDURE: 1. This problem is an indoor lab exercise and the required traverse data are given for the students to work on. There is no need to go to the field to observe and record traverse data since this is an exercise designed to develop within the student dexterity and familiarity in traverse computations. The lab instructor has the option to use the prepared data, revise it, or give a new set of data. 2. For the closed traverse data tabulated below, determine the following: a) Latitude and departure of each course. b) Adjusted latitudes and departures of each course (applying the compass rule). c) Area of the tract of land defined by the traverse (employing double meridian distances). d) Area of the same tract (employing double parallel distances). e) Linear error of closure and bearing of the side of error. LINE AB BC CD DE EA f)
LENGTH 639.35m 507.30m 678.70m 570.55m 1082.75m
AZIM FR SOUTH 170° 30’ 123° 05’ 56° 13’ 357° 58’ 270° 29’
Relative error of closure.
NOTE: It is advisable to plot the given traverse approximately to scale to better visualize the length and direction of each line, and the general shape of the given tract of land. 3. Tabulate given and computed values accordingly. Refer to the accompanying sample format for the tabulation of data.
LINE AB BC CD DE EA
s
LENGTH(m)
AZIMUTH FR. SOUTH
CALCULATED BEARING
LATITUDE +N -S
DEPARTURE +E -W
LINE
CORR
ADJ. LATITUDE +N -S
ADJ DEPARTURE +E -W
DMD
DOUBLE AREA
DPD
DOUBLE AREA
AB BC CD DE EA SUMS C. COMPUTATIONS: 1. Calculations required to determine the bearing of each line from the given azimuth will be obvious if a carefully prepared sketch of the given traverse is made. 2. The magnitude of the latitude of a line is the product of the length of the line by the cosine of its bearing angle. It is a positive value if the line bears towards the north and negative if towards the south. 3. The magnitude of the departure of a line is the product of the length of the line by the sine of its bearing angle. It is a positive value if the line bears towards the east and negative if towards the west. 4. The total correction or error of closure in latitude (C L) is the algebraic sum of all north and south latitudes. Similarly, the total correction of error of closure in departure (C D) is the algebraic sum of all east and west departures. 5. The latitudes and departures are adjusted as follows: a) The correction to be applied to the latitude of any line is to the total correction in latitude as the length of the line is to the length of the traverse or cl = (d/D) CL
Where: cl = correction to be applied to the latitude of any line. CL = total correction in latitude. d = length of the line. D = total length of the traverse.
b) If the sum of the north latitudes exceeds the sum of the south latitudes, a computed correction for a line is subtracted from the corresponding latitude if the line bears north and added if it bears south. Similarly, if the sum of south latitudes exceeds the sum of north latitudes, a computed correction for a line is added to the corresponding latitude if the line bears north and subtracted if it bears south. c) The correction to be applied to the departure of any line is to the total correction in departure as the length of the line is to the length of the traverse or cd = (d/D) CD Where: cd = correction to be applied to the departure of any line CD = total correction in departure D & d = some quantities as given above d) If the sum of the east departures exceed the sum of the west departures, a computed correction for a line is subtracted from the corresponding departure if the line bears east and added if bears west. Similarly, if the sum of the west departures exceeds the sum of east departures, a computed correction for a line is added to the corresponding departure if the line bears east and subtracted if it bears west. 6. The following are the rules in computing double meridian distances (DMDs). a) The DMD of the first line is equal to the departure of the line. b) The DMD of any other line is equal to the DMD of the proceeding line, plus the departure of the proceeding line, plus the departure of the line, plus the departure of the line itself. c) The DMD of the last line is numerically equal to the departure of the line but with opposite sign. This rule is used only as a check. 7. The following are the rules in computing double parallel distances (DPDs). a) The DPD of the first line is equal to the latitude of the line.
b) The DPD of any other line is equal to the DPD of the preceding line, plus the latitude of the proceeding line, plus the latitude of the line itself. c) The DPD of the last line is numerically equal to the latitude of the line but with opposite sign. This rule is used only as a check. 8. Double areas are determined by multiplying the DMD of the line by the adjusted latitude of the line. Double areas may either be positive or negative values. Similarly, double areas are also determined by multiplying the DPD of the line by the adjusted departure of the line. 9. The total area of the traverse is determined by dividing the algebraic sum of the double areas by two. The sign of the computed area is not significant. It may be a positive or a negative value. If the sign is negative, it is disregarded. 10. The linear error of closure is determined by computing the hypotenuse of the right angle triangle whose sides are the error of the latitude (C L) and the error of the closure in departure (CD) or LEC = √ (CL)^2 + (CD)^2 And similarly, the bearing of the side of error is computed by the following formula TanƟ = (-CD)/(-CL) For both equations, CD and CL are the same values determined in C(4). 11. The relative error of closure is determined by dividing the linear error of closure (LEC) by the sum of the individual lengths of the different lines of traverse of its perimeter. Express the numerator as one or unity and round off the resulting value in the denominator to the nearest hundred. D. REMARKS, HINTS & PRECAUTIONS 1. Before the DMD or DPD method can be applied properly, the latitudes and departures of the traverse should first be adjusted such that the figure becomes a closed polygon. This means that the algebraic sum of the north and south latitudes is zero, and the algebraic sum of the east and west departures is also zero. 2. The compass rule is suitable for surveys where the angles and distances are measured with equal precision. It is most commonly used in actual practice. When the rule is applied, it is assumed that all lines were measured with equal care, and all angles taken with the same precision themselves. 3. The transit rule is also used for adjusting traverses. It is theoretically better for surveys where the angles are measured with greater accuracy than the distances such as in stadia surveys. 4. The correctness of the area computation by the DMD method can be readily checked by applying the double parallel distance method. The DPD method is essentially the same as the DMD method and the only difference is that the lines are projected upon the reference parallel instead of upon the reference meridian.
LAB EXERCISE NO.8 PROBLEM: DETERMINING AREA OF A FIELD WITH IRREGULAR/CURVED BOUNDARY OBJECTIVE: To learn how to determine the area of a tract of land with an irregular or curved boundary. LAB EXERCISE OUTLINE A. INSTRUMENTS & ACCESSORIES: Steel Tape, Pegs or Chaining Pins, and Range Poles. B. PROCEDURE: 1. Set two pegs or markers on both ends of the designated base (or reference) traverse line located near an irregular or curve boundary. Call these points A and B. 2. Measure its horizontal length twice and record the mean as the actual length of the line. 3. Divide the reference line into and even number of intervals regularly spaces not more than 5.0 meters long and use pegs to mark each division along the line. 4. At each point established, erect a perpendicular to the reference line with the use of a steel tape, and where the perpendicular intersects the irregular or curved boundary set another peg on the ground. 5. Measure the horizontal distance from the peg on the regerece line to its corresponding peg established on the boundary line. Call these short measured distances (offsets) as h1, h2, h3, and etc. 6. Tabulate all measured values accordingly. Refer to the accompanying sample format for the tabulations of field data. Length of Base Line:______________ Interval Between Offsets: ______________
OFFSETS
MEASURED LENGTH (m)
OFFSETS
MEASURED LENGTH (m)
C. COMPUTATIONS: Calculate the area of the tract using the following: 1. Area by Trapezoidal Rule. When using the trapezoidal rule, it is assumed that the boundary is composed of chords connecting the ends of the offsets as shown in the figure below.
The area bounded by the irregular boundary, the reference line and the ends offsets is determined as follows. A=d(h1+ hn + h2 + h3 + … hn-1) 2 Where : A = the total area or summation of the areas of the trapezoids comprising the total area d = common spacing of the offsets n = number of offsets h1 = the first offset, and h 2, h3, and etc. are the intermediate offsets hn = the last offset 2. Area by Simpson’s One-Third Rule. This rule assumes that the curve through each successive three points is a portion of a parabo la as shown in the figure below.
The area bounded by the curved boundary, the reference line and the end offsets is determined as follows A=d [( h1+ hn ) + 2 ( h3 + h5 + … hn-2) + 4(h2 + h4+ … hn-1) 3 Where : A = the total area of the field with a curved boundary d = common spacing of the offsets n = number of offsets h1 = the first offset, and h 2, h3, and etc. are the intermediate offsets hn = the last offset
NOTE: The above formula shows that the sum of all odd numbered offsets is multiplied twice, and the sum of all even numbered offsets is multiplied four times. D. REMARKS, HINTS, & PRECAUTIONS: 1. In the trapezoidal rule, the ends of the offsets in the boundary line are assumed to be connected by straight lines thereby, forming a series of trapezoids, the bases being the offsets and the altitudes being the common distance between offsets. 2. When applying the trapezoidal rule, no considerable error is introduced when the offsets are taken close enough together and when the boundary consists of flat curves. 3. If the boundaries are found to be curved, Simpson’s one-third rule is considered better to use than the trapezoidal rule The rule assumes that the curve through each successive three points is a portion of a parabola. It also assumes that the offsets are e qually and closely spaced. 4. Results obtained by using Simpson’s one-third rule are greater or smaller than those obtained b y using the trapezoidal rule, depending upon whether the boundary curve it concave or convex toward the traverse line. 5. Simpson’s one-third rule is applicable only if there is an odd number of offsets. If the total number or offsets is even, the partial area at either end o f the series of offsets is computer separately in order to make n for the remaining area an off number and thus making the rule applicable. 6. Areas computed by the transit rule or Simpson’s one-third rule only yield approximate values.
