The Mathematics of Engineering Surveying (3)

Transcription

1 The Mathematics of Engineering Surveing (3) Scenario s a new graduate ou have gained emploment as a graduate engineer working for a major contractor that emplos 2000 staff and has an annual turnover of 600m. s part of our initial training period the compan placed ou in their engineering surveing department for a si-month period to gain eperience of all aspects of engineering surveing. One of our first tasks was to work with a senior engineering surveor to establish a framework of control surve points for a new 2m highwa development consisting of a two mile b-pass around a small rural village that, for man ears, has been blighted b heav traffic passing through its narrow main street. Having established the control framework ou are now required to establish the position of a number of additional control points to be subsequentl used to establish the road centre-line. In this eercise ou will carr out the geometric calculations that would enable ou to determine the precise position of the new control points using the coordinates of the eisting control surve points and surve measurements. The technique that ou will use is referred to as a resection technique Importance of Eemplar in Real Life When surveing on the civil engineering or construction site it is often necessar to find the coordinates of new control points or points of detail. This is relativel simple if both the eisting and the new point are accessible but often one of them is not and so other techniques are required. For eample the new or eisting surve points ma be targets on walls, points on high buildings or points on land to which access is denied. When the eisting points are accessible but the new point is not then the surve needs to emplo an intersection technique. When the eisting points are not accessible but the new point is then the surve needs to emplo a resection technique. Figures and 2 show the road construction scheme where the control surve points will have been established along the approimate line of the road using a traverse technique. The pronounced circular curves of the road can be clearl seen and for each curve it will be necessar to establish the eact position of the curve s centre-line points so that the full curve alignment can be accuratel established. To facilitate this it ma be necessar to establish a number of additional control points nearer to the road centre-line than the originall established control framework. The techniques described in this eemplar might well be used for this purpose. Figure : Road alignment as seen on a map Figure 2 Road alignment as seen from the air ackground Theor Two alternative resection methods are described below: --

2 Method (a) In figure 3 below the point is the location of a new control station that is to be established using a resection technique from the three established traverse stations or points of known coordinates,, and should be visible from as shown in figure 3. In the figure,, and are known fied points, from which the distances and can be calculated. The angle ϕ can also be calculated from the coordinates of, and and is hence a known quantit. The angles α and β will be measured b accurate sightings to the three traverse stations in the field and will hence be known, whilst the angle θ has to be calculated. Once θ has been calculated the coordinates of can then be determined. orth φ 360-(α+β+θ+φ) θ α β Figure 3: Establishing b Resection - Method (a) In the above figure the angle at = 360 ( α + β + θ + φ) = (360 α β φ) θ = X θ where X contains site-measured ( α, β ) or readil calculated (φ ) angles and θ is an unknown angle, to be determined. From triangle and using the sine rule: = sinα sin( X θ ) From triangle and using the sine rule: = sin β Therefore: ) = = sinα sin β ) sin β = = K sinα () Where: sin β K = (2) sinα K is fied value as it can be calculated from the two known fied distances (, ) and two measured angles ( α, β ). Therefore from equation (): Hence: ) = K sin X cosθ cos X = K sin X cotθ cos X = K K + cos X cotθ = (3) sin X -2-

3 Equation (3) can be used to calculate the angle θ. Once this angle is known then the geometr of the triangles can be solved to determine the distances, and. For eample appling the sine rule to triangle : sin( 80 θ α) = (4) sin α and, knowing the coordinates of station, the coordinates of can hence be calculated from the length of and the bearing of (see the eemplar The Mathematics of Engineering Surveing () for this part of the calculation). Onl one distance, or and their corresponding bearings are needed to calculate the coordinates of but if one is used then the others can then be used as a check on the accurac of the computation. Method (b) We will not give the proof for the geometric relationships given below but Method (b), known as Tienstra s Method provides an alternative and possibl neater solution to that given above. Figure 4 again shows the three points of known coordinate, and, and the fourth point,, the coordinates of which are to be established b resection. The three angles ( ), ( ) and ( ) are measured in the field and are measured in a clockwise direction as shown. Three separate diagrams are shown in figure 4 to take into account three possible positions of in relation to, and. Once these angles are measured in the field the coordinates of (E p and p ) can be calculated using the equations (5),(6) and (7) given below (a) ++ = 360 (a) ++ = 360 (a) ++ = 720 Figure 4: Establishing b Resection - Method (b) alculate: a = tan tan E b = tan tan (5) c = tan then calculate: K = cot a cot K 2 tan = cot b cot hence, calculate the Easting and orthing: K 3 = (6) cot c cot -3-

4 E KE + K 2E + K3E = and K + K + K 2 3 K + K + K 2 3 = (7) K + K 2 + K3 It should be noted that method (b) can not be used if the three known points, and lie on a straight line and neither method can be used if all four points lie on the circumference of a circle. Questions Eample [Method (a)] The table below gives the coordinates of three of the traverse points established for a section of the new road. further control station is to be established using the resection technique and b sighting on to stations, and. The angles measured in the field are = 40 o ' 24 '' 08 and = 57 o 36 ' 00 ''. alculate the angle θ, the length and hence the coordinates of this newl established control station. (refer to figure 3) Eample [Method (b)] Using the same data as above calculate the coordinates of using Tienstra s Method. The clockwise angles measured in the field are: = = ' 00 '' 57 o 36 ' 36 '', = = 262 o 5, = = ' 24 '' 40 o 08 Station Easting orthing (metres) (metres) Where to find more.. Schofield W, reach M, Engineering Surveing, 6 th edn, Oford: utterworth-heinemann 2. ird J, Engineering Mathematics, 5 th edn, Elsevier, 2007 (IS ) ooooo -4-

5 The Mathematics of Engineering Surveing (3) IFORMTIO FOR TEHERS Teachers will need to understand and eplain the theor outlined above and have knowledge of: Some terminolog relating to engineering surveing Geometr and trigonometr Topics covered from Mathematics for Engineers Topic : Mathematical Models in Engineering Topic 3: Models of Oscillations Topic 5: Geometr Learning Outcomes LO 0: Understand the idea of mathematical modelling LO 03: Understand the use of trigonometr to model situations involving oscillations LO 05: Know how 2-D and 3-D coordinate geometr is used to describe lines, planes and conic sections within engineering design and analsis LO 09: onstruct rigorous mathematical arguments and proofs in engineering contet LO 0: omprehend translations of common realistic engineering contets into mathematics ssessment riteria.: State assumptions made in establishing a specific mathematical model.2: Describe and use the modelling ccle 3.: Solve problems in engineering requiring knowledge of trigonometric functions 5.: Use equations of straight lines, circles, conic sections, and planes 5.2: alculate distances 9.: Use precise statements, logical deduction and inference 9.2: Manipulate mathematical epressions 9.3: onstruct etended arguments to handle substantial problems 0.: Read criticall and comprehend longer mathematical arguments or eamples of applications Links to other units of the dvanced Diploma in onstruction & The uilt Environment Unit 2 Unit 3 Unit 6 Site Surveing ivil Engineering onstruction Setting Out rocesses Solution to the Questions ' 36 '' Method [a]: θ = 39 o 03, = 68.30m, E =67.446m, = m Method [b]: E =67.446m, = m These eercises can be replicated with other sets of coordinates. However the geometr can be quite challenging and the learner should sketch out each problem, ideall on graph paper to visualise the problem and its solution. ooooo -5-