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errors in taping

When a tape is suspended between two measuring heads . And B  , both at same level , the shape it takes up is a catenary . if C is the lowest point on the curve , then on length CB there are three forces acting , namely the tension T at B to at C and the weight of portion CB ,

where is the weight per unit length and s is the are length CB thus must be in equilibrium under the action of these three forces hence .










Slope :

if the difference in height of the two measuring heads is h , the slope distance L andhorizontal equivalent D , then by pythagoras :


D  =  (L ² - h ²) 1/2

Alternatively if the vertical angle of the slope of the ground is measured then :

D = L cos ⦵

and the correction C ⦵  = L - D 

C ⦵ = L ( 1 - cos ⦵ )


Altitude :

if the surveys are to be connected to the national mapping system of country , the distance will need to be reduces to the common datum of that system , namely MSL Alternatively , if the engineering scheme is of a local nature . 

distances may be reduced to the mean level of the area . This has the advantage setting -out distances on the ground are .  without sensible error , equal to distance computed from coordinates in the mean datum plane .

Consider in whch a distanced in a plane situated at a height H above MSL .

                                                    R
By similar triangles   M =  ___________  ×  L
                                               R + H

                                                           RL                                  R                           LH
∴  Correction  Cm = L - M  = L  -  _______   =  L  ( 1 -  _________  )  =  ___________
                                                           R+H                            R + H                    R + H

As H is normally negligible compared with R in the denominator 

                LH
CM  =  ______
                  R





The corrction is negative for surface work above MSL or mean level of the area but may be positive for tunnelling or mining work below MSL or mean level of the area .






sag



step measurement

step measurement is the process of breaking the overall distance down into manageable short sections each much less than a whole tape length . The tape is stretched horizontally and a plumb-bob suspended from the elevated end of the tape . this method of measurement over sloping ground should be avoided if hogh accuracy is required 

The main source of error lies in attempting to accurately locate the suspended end of the tape .

The steps should be kept short enough to minimize sag in the tape , Thus the sum of steps equals the horizontal distance required .


To eliminate or minimize the systematic errors of taping , it is necessary to adjust each measured to its final horizontal equivalent as follows .

STANDARDIZATION :

During a period of  use , a tape will gradually alter in length for a variety of reasons . The amount of change can be found by having the tape  standardized at either the National physical Laboratory (NPL)  for invar tapes or the Department of Trade end industry (DTI) for stel tapes , or by comparing it with a 70 N tension , or as 30 m exactly at a temperature other standard .


WORKED EXAMPLES :

Example 4.1  A distance of 220.450 m was measured with a steel band of nominal legth 30 m . On  standardization the tape was found to be 30.003 m . Calculate the correct measured distance , assuming the error is evely distributed throughout the tape .


Error per 30 m =  3 mm

                                                     220.450
 ∴Correction for total length =  (  _________ )   ×  3 mm  =  22  mm
                                                           30


∴ Correct length is 220.450 +  0.022  =   220,472  m


Note that :

(1)  shows that when the tape is too long , the distance measured appears too short , and the correction is therefore positive . The reverse is the case when the tape is too short .


(2)  When  setting out a distance with a tape the rule in (1) are reversed .

(3) It  is better to compute Example 4.1 on the basic of the correction (as shown ) , rather than the total corrected length .
in this way fewer significant figures are required .













Temperature 

Tapes are usually standardized at 20ًC Any varion above or below this value will cause the tape to expand or contract , givivg rise to systematic errors . Difficulty of obtaining true temperatures of the tapes lead to the use invar tapes . Invar is a nickel - steel alloy with a very low coefficient of expansion .

Coefficient of expansion of steel  K = 11.2 × 10⁻₆  per ُC

Coefficient of expansion of invar  K = 0.5 × 10⁻⁶ Per ُC

Temperature correction    Ct = KL ⋴⧍ t

where ∆ t  = difference between the standard  and field temperatures ( ُC) = ( ts - t a)

The sign of the correction is in accordance with the rule specified in note (1) above .


Tension 

Generally the tape is used under standard tension , in which case there is no correction . It may , however be necessary in certain instances to apply a tension greater than standard . from Hook's law :

stress = stain × a constant 


This constant is same for a given material and is called the modulus of elasticity (E) since strain is a non-dimensional quanitity , E has the same dimensions as strees , i ,e N/mm₂

                  ⩟ T
∴ Ct = L × ____
                AE

⩟ T  is normally the total acting on the cross-section , but as the tape would be standardized under tension . ⩟T in this case is the amount of stress greater than standard . Therefore ⩟T   is the difference between field and standard tension , This value may be measured in the field in kilograms and should be converted to newtons (N) for compatiblity with the other units used in the formula ,i,e  1 kgf  = 9.80665 N





E is modulus of elasticity in N/mm₂ : L is measured lenfth in m : and Ct is the extension and thus correction to the tape length in m . As the tape is stretched under the extra tension . the correction is positive .





distance adjustment

the most accurate way to measure distance with a steel is to measure the distance between pre-set measuring marks rather than attempt to mark the end of each tape length the procedure is as follows :

(1) The survey points to be measured are defined by nails in pegs should be set flush with the ground surface . Ranging rods are then set behind each peg , in the line of measurement .




(2) Using a linen tape , arrows are aligned between the two points at intervals less than a tape length measuring plates are than set firmly in the ground at these points , with their measuring edge normal to the direction of taping .

(3) The steel band is then carefully laid out , in a straight line between the survey point and the first plate . One end of the tape is firmly anchored , whilst tension is slowly applied at the other end .At the exact instant of standard tension ,both ends of the tape are read simultaneously against the survey station point and the measuring plate edge respectively ,on command from the person applying the tension the tension is eased and the whole process repeated at least four times or until a good set of results is obtained .

(4) When reading the tape , the meteres , decimetres and centimeteres should be noted as the tension is being applied thus on the command , to read only the millimeters are required.


(5) The reading are noted by the booker and quikly subtracted from each other to give the length of the measured bay .

(6) In addition to rear and fore readings ,the tape temoerature is recorded , the value of the applied tension , which may in some instances be greater than standard , and the slope or difference in level of the tape ends are also recorded .

(7) This method requires a survey party of five : one to anchor the tape end , one to apply tension , two observers to read the tape and one booker .

(8) The process is repeated for each bay of the line being measured , care being taken not to move the first measuring plate , which is the start of the second bay , and so on .

(9) The data may be booked as follows :




The mean result is then corrected for :

(1) Tape standardization .

(2) Temperature .

(3) Tension (if necessary)

(4) Slope

The final total distance may be reduced to its equivalent MSL or mean site level.






measuring along the ground




Taapes come in a variety of lengths and materials . for engineering work the lengths are generally 10 m , 30 m , 50 m and 100 m .

Linen or glass fibre tapes may be used for general use , where precision is not a prime consideration the linen tapes are made from high quality linen , combined with metal fibres to increase their strength .

They are sometimes encased in plastic boxes with recessed handles . These tapes are often graduated in 5-mm intervals only .

 More precise versions of the above tapes are made of steel and graduated in millimeters .

For high-accuracy work , steel bands mounted in an open frame are used . They are standardized so that they measure their nominal length at a designated temperature usually 20ً C and at a designated applied  tension usually between 50 N to 80 N .

This information is clearly printed on the zero end of the tape .shows a sample of the equipment .

For the most precise work , invar tapes made from 35% nickel and 65 % steel are available . The singular advantage of such tapes is that they have a negligible coefficient of expansion compared with steel ,


and hence temperature variation are not critical , Their disadvantages are that the metal is soft and weak , whilst the price is more than ten times that of steel tapes . An alternative tape , called a lovar tape , is roughly , midway between steel and invar .

Much ancillary equipment is necessary in the actual taping 

process , e.g
(1) Ranging rods are made of wood or steel , 2 m long and 25 mm in diameter painted alternately red and white and have pointed metal shoes to allow them to be thrust into the ground . They are generally used to align a straight line between two points .

(2) Chaining arrows made from No. 12 steel wire are used to mark the tape lengths .








(3) Spring balances are generally used with roller-grips or tapeclamps to grip the tape firmly when the standard tension is applied .

As it is quite difficult to maintain the exact tension required with a spring balance , it may be replaced by a tension handle , which ensures the application of correct tension .


(4) Field thermometers are also necessary to record the tape temperature at the time of measurement , thereby permitting the computation of tape corrections when the temperature varies from standard these thermometers are metal cased and can be clipped onto the tape if necessary , or simply laid on the ground alongside the tape but must be shaded from the direct rays of the sun .

(5) Hand levels may be used to ensure that the tape is horizontal . This is basically a hand-held tube incorporation a spirit bubble to ensure a horizontal line of sight .Alternatively .an Abney level may be used to measure the slope of the ground .
(6) plumb-bobs may be necessary if stepped taping is used.

(7) Measuring plates are necessary in rough ground , to afford a mark against which the tape may be read. shows the tensioned tape being read against the edge of such a plate. The corners of the triangular plate are turned down to form grips .

when the plate is pressed into the earth and thereby prevent its movement .

In addition to the above , light oil and cleaning rags should always be available to clean and oil the tape after use .







tapes

distance measurment

The ease with which total stations produce horizontal distance vertical height and horizontal direction makes them ideal instruments for rapid and accurate contouring in virtually any type of terrain the data recorded may be transformed from direction 

distance and elevation of point to its position and elevation in terms of three-dimensional coordinates .These thus comprise a digital terrain or ground model (DTM/DGM) from which the contours are interpolated and plotted .

the total station and a vertical rod that carries a single reflector are used to locate the ground points ,A careful reconnaissance  of the area is necessary , in order to plan the survey and define the necessary ground points that are required to represent the characteristic shape of the terrain .

Break lines , the tops and bottoms of hills or depressions , the necessary features of water courses , etc ..

and enough points to permit accurate interpolation of contour lines at the interval required . comprise the field data .
As the observation distances are relatively short , curvature and refraction might be ignored , However in most total stations corrections for curvature and refraction might be ignored However in most total station corrections for curvature and refraction may be applied .









it can be seen that if the reduced level of point A (RLa) is known then the reduced level of ground point B is :

RLb  = RLa + hi + ∆ h - ht


When contouring the height of the reflector is set to the same height as the instrument , ie  ht  = hi  , and cancels out in the previous equation , Thus the height displayed by the insttrument is the height of the ground point above A :

RLb  =  RLa  +  ∆ h

in this way the reduced levels of all the ground points are rapidly acquired and all that is needed are their position , One method of carrying out the process is by radiation .

the instrument is set up on a control point A , whose reduced level is known and sighted to a second control point (RO) the horizontal circle is set to the direction computed from the  coordinate of A and the RO . 

The instrument is then turned through a chosen horizontal angle (∅) defining the direction of the first ray . Terrain points along this ray are then located by measured horizontal distance and height difference . This process is repeated along further rays until the area is covered 


unless a very experienced person is used to locate the ground points , there will obviously be a greater density of points near the instrument station . the method however is quite easy to organize in the field . the angle between Many ground - modelling software package interpolate and plot contours from strings of linked terrain points .

computer processing is aided if the ground points are located in continuous strings throughout the area , approximately following the line of the contour , they may also follow the line of existing watercourse , roads , hedges , kerbs , etc , 


Depending on the software package used , the string points may be transformed into a triangular or gridded structure . Heights can then be determined by linear interpolation and the terrain represented by simple planar triangular facets . Alternatively , high -order polynomials may be used to define three-dimensional surfaces fitted to the terrain points . from these data , contours are interpolated amd a contour model of the terrain produced .





contouring

Consider the formula for a single observation found by substituting equations 3.8 into 3.13: 

∆ H   =  S sin ∝ + hi  +  ht  +  S2  (1 - K) / 2R

The obvious sources of error lie in obtaining the slope distance S , the vertical angle ∝ the heights of the  instrument and target , the coefficient of refraction K and a value for the local radius of the Earth R . Differentiating gives :




Once again it can be seen that the accuracy to measure S  is not critical .
However , the measurement of the vertical angle is critical and the importance of its precision will 
increase with greater distance , The error  in the value of refraction is the most critical component and will increase rapidly as the square of the distance . Thus to achieve reasonable results over long sights simultaneous reciprocal observations are essential .




sources of error



Recipro observations are observations from A and B the arithmetic mean result being accepted if one assumes a symmetrical line of sight from each and the observations are taken simultaneously 

Then the effect of curvature and refraction is cancelled out for instance for elevated sights (c-r) is added to a positive value to increase the height difference , for depressed or downhill sights (c-r) is addes to a negative of curvature and refraction . this statement is not entirely true as the assumption of symmetrical line of sight from each is dependent on uniform ground and atmosphereic conditions at end , at the instant of simultaneus observation .

In practice over short distance , sighting into each other's object lens form an excellent target with some form of communication to ensure simultaneous observation.

The  following numerical example is taken from an actual survey in which the elevation of A and B had been obtained by precise geodetic levelling and was checked by simultaneous reciprocal trigonometrical levelling .


WORKED EXAMPLE 






As the observations are reciprocal the corrections for curvature and refraction are ignored :




This value compares favourably with 2.311 m obtained by precise levelling , However , the disparity between the two values 0.846 and - 3.722 shows the danger inherent in single observations uncorrected for s single for a single observation .

from A to B 

2.284  =  4279.446 cos   89ُ  59 َ   18,7 ً  +  1.290  - 1.300 + (c   -  r  )

where ( c - r )  =  S2  ( 1  - K  ) 2 R 

and the local value of R for the area of observation  =  6 364  700 m

2.284  =  0.856  -  0.010  +  S2  (1 - K ) / 2R 

1.438  =  4279.4462 (1  -  K  ) / 2  ×  6 364 700 m

K  =  0.0006


From B to A :

2.284  =  - 3.732  +  1.300   -  1.290  +  S2  (1  - K ) / 2R

K =  0.0006


Now this value for K could be used for single ended observations taken within the same area , at the same time , to give improved results .

A variety of formulae are available for finding K directly . For example , using zenith angles :

                     Za  +  Zb  -  180ُ             R
 K  =  1  -  ______________     ×   _____
                             180ُ / ∩                    S



and using vertical angles :


K =  ( ⊖  +  ∝ o +  B o ) / ⊖


where ⊝ ًً  =  Sp/R   where P = 206265

In the above formulae the values used for the angles must be those which would have been observation had 

hi  =  ht  and , in Case of  vertical angles , entered  with their appropriate  sign  .





By sine rule :  ∝ - e    and  for an angle of depression it becomes bo  = B + e .

                      ht  - i sin (90ُ  -  ∝  )
sin   e  =   ___________________
                                  S

                                        ht  -  i  cos ∝
        e   =    sin  - 1   (  ______________  )
                                                 S

                         ht  - 1                          ht3  - i
              =   _______  cos    ∝   +    ________  cos  3  ∝  + ............
                          S                                 6S3



∴   e   =  (  ht  -  i cos  ∝ ) / S


For  zenith  angles :


e  =  ( ht  -  i  sin Z ) / S


reciprocal observations