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### Surveying Problems and Solutions Manual

Surveying Problems and Solutions Manual

675 Pages

F.A. Shepard

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Fundamentals of Surveying Theory and Samples Exercises

This book is an attempt to deal with the basic mathematical aspects of 'Engineering Surveying', i.e. surveying applied to construction and mining engineering projects, and to give guidance on practical methods of solving the typical problems posed in practice and, in theory, by the various examining bodies.

The general approach adopted is to give a theoretical analysis of each topic, followed by worked examples and, finally, selected exercises for private study. Little claim is made to new ideas, as the ground covered is elementary and generally well accepted. It is hoped that the mathematics of surveying, which so often causes trouble to beginners, is presented in as clear and readily understood a manner as possible. The main part of the work of the engineering surveyor, civil and mining engineer, and all workers in the construction industry is confined to plane surveying, and this book is similarly restricted.

TOC

1 LINEAR MEASUREMENT 1

1.1 The basic principles of surveying 1

1.2 General theory of measurement 2

1.3 Significant figures in measurement and computation 3

1.4 Chain surveying 6

1.41 Corrections to ground measurements 6

1.42 The maximum length of offsets from chain lines 13

1.43 Setting out a right angle by chain 15

1.44 To find the point on the chain line which produces a perpendicular from a point outside the line 16

1.45 Obstacles in chain surveying 17 Exercises 1(a) 22

1.5 Corrections to be applied to measured lengths 23

1.51 Standardization 23

1.52 Correction for slope 23

1.53 Correction for temperature 26

1.54 Correction for tension 27

1.55 Correction for sag 32

1.56 Reduction to mean sea level 38

1.57 Reduction of ground length to grid length 39

1.6 The effect of errors in linear measurement 45

1.61 Standardization 45

1.62 Mal alignment and deformation of the tape 45

1.63 Reading or marking the tape 46

1.64 Errors due to wrongly recorded temperature 46

1.65 Errors due to variation from the recorded value of tension 47

1.66 Errors from sag 48

1.67 Inaccurate reduction to the horizontal 51

1.68 Errors in reduction from height above or below mean sea level 52

1.69 Errors due to the difference between ground and grid distances 52 Exercises 1(b) 52

2 SURVEYING TRIGONOMETRY 57

2.1 Angular measurement 57

2.11 The degree system 57

2.12 Trigonometrical ratios 58

2.13 Complementary angles 60

2.14 Supplementary angles 60

2.15 Basis of tables of trigonometrical functions 63

2.16 Trigonometric ratios of common angles 64

2.17 Points of the compass 65

2.18 Easy problems based on the solution of the right angled triangle 67 Exercises 2(a) 71

2.2 Circular measure 72

2.21 The radian 72

2.22 Small angles and approximations 73

2.3 Trigonometrical ratios of the sums and differences of two angles 77

2.4 Transformation of products and sums 79

2.5 The solution of triangles 80

2.51 Sine rule 80

2.52 Cosine rule 81

2.53 Area of a triangle 82

2.54 Half-angle formulae 82

2.55 Napier's tangent rule 83

2.56 Problems involving the solution of triangles 83

2.6 Heights and distances 91

2.61 To find the height of an object having a vertical face 91

2.62 To find the height of an object when its base is inaccessible 92

2.63 To find the height of an object above the ground when its base and top are visible but not

accessible 95

2.64 To find the length of an inclined object on the top of a building 98

2.65 To find the height of an object from three angles of elevation only 100

2.66 The broken base line problem 102

2.67 To find the relationship between angles in the horizontal and inclined planes 106 Exercises 2(b) 108

CO-ORDINATES 112

3.1 Polar co-ordinates 112

3.11 Plotting to scale 113

3.12 Conversion of the scales 113

3.13 Scales in common use 114

3.14 Plotting accuracy 114

3.15 Incorrect scale problems 114

3.2 Bearings 115

3.21 True north 115

3.22 Magnetic north 115

3.23 Grid north 116

3.24 Arbitrary north 116

3.25 Types of bearing 117

3.26 Conversion of horizontal angles into bearings 121

3.27 Deflection angles 124 Exercises 3(a) 126

3.3 Rectangular co-ordinates 127

3.31 Partial co-ordinates, AE, AN 128

3.32 Total co-ordinates 128 Exercises 3(b) (Plotting) 131

3.4 Computation processes 133

3.41 Computation by logarithms 134

3.42 Computation by machine 134

3.43 Tabulation process 135

3.44 To obtain the bearing and distance between two

points given their co-ordinates 136

3.5 To find the co-ordinates of the intersection of two lines 146

3.51 Given their bearings from two known co-ordinate

stations 146

3.52 Given the length and bearing of a line AB and all the angles A, B and C 149 Exercises 3(c) (Boundaries) 157

3.6 Transposition of grid 158

3.7 The National Grid Reference system 160

Exercises 3(d) (Co-ordinates) 163 Appendix (Comparison of Scales) 169

4 INSTRUMENTAL OPTICS 170

4.1 Reflection at plane surfaces 170

4.11 Laws of reflection 170

4.12 Deviation by successive reflections on two inclined mirrors 170

4.13 The optical square 171

4.14 Deviation by rotating the mirror 171

4.15 Principles of the sextant 172

4.16 Use of the true horizon 174

4.17 Artificial horizon 175

4.18 Images in plane mirrors 176

4.19 Virtual and real images 177

4.2 Refraction at plane surfaces 177

4.21 Laws of refraction 177

4.22 Total internal reflection 177

4.23 Relationships between refractive indices 178

4.24 Refraction through triangular prisms 179

4.25 Instruments using refraction through prisms 180 Exercises 4(a) 184

4.3 Spherical mirrors 184

4.31 Concave or converging mirrors 184

4.32 Convex or diverging mirrors 186

4.33 The relationship between object and image in curved mirrors 186

4.34 Sign convention lg7

4.35 Derivation of formulae Igg

4.36 Magnification in spherical mirrors 190

4.4 Refraction through thin lenses 191

4.41 Definitions 191

4.42 Formation of images 192

4.43 The relationship between object and image in a thin lens 193

4.44 Derivation of formulae 193

4.45 Magnification in thin lenses 195

4.5 Telescopes 196

4.51 Kepler's astronomical telescope 196

4.52 Galileo's telescope 196

4.53 Eyepieces I97

4.54 The internal focusing telescope 198

4.55 The tachometric telescope (external focusing) 201

4.56 The anallatic lens 203

4.57 The tachometric telescope (internal focusing) 207

4.6 Instrumental errors in the theodolite 210

4.61 Eccentricity of the horizontal circle 210

4.62 The line of collimation not perpendicular to the trunnion axis 213

4.63 The trunnion axis not perpendicular to the vertical axis 215

4.64 Vertical axis not truly vertical 217

4-65 Vertical circle index error 219

4.7 The auxiliary telescope 228

4.71 Side telescope 228

4.72 Top telescope 233

4.8 Angular error due to defective centering of the theodolite 234

4.9 The vernier 237

4.91 Direct reading vernier 237

4.92 Retrograde vernier 238

4.93 Special forms used in vernier theodolites 238

4-94 Geometrical construction of the vernier scale 238 Exercises 4(b) 240

LEVELLING 244

5. 1 Definitions 244

5.2 Principles 245

5.3 Booking, of readings 246

5.31 Method 1, rise and fall 246

5.32 Method 2, height of collimation 247 Exercises 5 (a) (Booking) 254

5.4 Field testing of the level 257

5.41 Reciprocal levelling method 257

5.42 Two-peg method 259 Exercises 5 (b) (Adjustment) 264

5.5 Sensitivity of the bubble tube 267

5.51 Field test 267

5.52 O-E correction 268

5.53 Bubble scale correction 268 Exercises 5(c) (Sensitivity) 270

5.54 Gradient screws (tilting mechanism) 271

5.6 The effect of the earth's curvature and atmospheric

refraction 272

5.61 The earth's curvature 272

5-62 Atmospheric refraction 273

5.63 The combined effect of curvature and refraction 273 Exercises 5(d) (Curvature and refraction) 275

5.64 Intervisibility 275 Exercises 5 (e) (Intervisibility) 277

5.65 Trigonometrical levelling 278

5.7 Reciprocal levelling 279

5.71 The use of two instruments 281 Exercises 5(f) (Reciprocal levelling) 282

5.8 Levelling for construction 283

5.81 Grading of constructions 283

5.82 The use of sight rails and boning (or travelling) rods 284

5.83 The setting of slope stakes 286

Exercises 5(g) (Construction levelling) 288

Exercises 5 (h) (General) 289

6 TRAVERSE SURVEYS 298

6. 1 Types of traverse 298

6.11 Open 298

6.12 Closed 298

6.2 Methods of traversing 299

6.21 Compass traversing 300

6.22 Continuous azimuth method 301

6.23 Direction method 302

6. 24 Separate angular measurement 304 Exercises 6(a) 304

6.3 Office tests for locating mistakes in traversing 306

6.31 A mistake in the linear value of one line 306

6.32 A mistake in the angular value at one station 307

6.33 When the traverse is closed on to fixed points and a mistake in the bearing is known to exist 307

6.4 Omitted measurements in closed traverses 308

6.41 Where the bearing of one line is missing 308

6.42 Where the length of one line is missing 309

6-43 Where the length and bearing of a line

are missing 309

6.44 Where the bearings of two lines are missing 309

6.45 Where two lengths are missing 314

6.46 Where the length of one line and the bearing of another line are missing 315 Exercises 6(b) (Omitted values) 316

6.5 The adjustment of closed traverses 317

6.51 Where the start and finish of a traverse are fixed 317

6-52 Traverses which return to their starting point 323

6.53 Adjusting the lengths without altering the bearings 323

6.54 Adjustment to the length and bearing 330

6.55 Comparison of methods of adjustment 336

Exercises 6 (c) (Traverse adjustment) 348

Exercises 6(d) (General) 352

7 TACHEOMETRY 359

7.1 Stadia systems — fixed stadia 359

7.2 Determination of the tacheometric constants m and K 360

7.21 By physical measurement of the instrument 360

7.22 By field measurement 361

7.3 Inclined sights 362

7.31 Staff normal to the line of sight 362

7.32 Staff vertical 363

7.4 The effect of errors in stadia tacheometry 367

7.41 Staff tilted from the normal 367

7.42 Error in the angle of elevation with the staff normal 367

7.43 Staff tilted from the vertical 368

7.44 Accuracy of the vertical angle to conform to the overall accuracy 371

7.45 The effect of the stadia intercept assumption 372 Exercises 7(a) 380

7.5 Subtense systems 383

7.51 Tangential method 383

7.52 Horizontal subtense bar system 388

7.6 Methods used in the field 392

7.61 Serial measurement 392

7.62 Auxiliary base measurement 393

7.63 Central auxiliary base 395

7.64 Auxiliary base perpendicularly bisected by the traverse line 397

7.65 Two auxiliary bases 398

7-66 The auxiliary base used in between two traverse lines 400

Exercises 7(b) 403

8 DIP AND FAULT PROBLEMS 411

8.1 Definitions 411

8.2 Dip problems 413

8.21 Given the rate and direction of full dip, to find the apparent dip in any other direction 413

8.22 Given the direction of full dip and the rate and direction of an apparent dip, to find the rate of full dip 413

8.23 Given the rate and direction of full dip, to find the bearing of an apparent dip 415

8.24 Given two apparent dips, to find the rate and direction of full dip 416

8.25 Given the rate of full dip and the rate and direction of an apparent dip, to find the direction of full dip 421

8.26 Given the levels and relative positions of three points in a plane (bed or seam), to find the direction and rate of full dip 422

8.3 Problems in which the inclinations are expressed as angles and a graphical solution is required 427

8.31 Given the inclination and direction of full dip, to find the rate of apparent dip in a given direction 427

8.32 Given the inclination and direction of full dip, to find the direction of a given apparent dip 428

8.33 Given the inclination and direction of two apparent dips, to find the inclination and direction of full dip 429

Exercises 8(a) 429

8.4 The rate of approach method for convergent lines 432

8.5 Fault problems 437

8.51 Definitions 437

8.52 To find the relationship between the true and apparent bearings of a fault 443

8.53 To find the true bearing of a fault when the throw of the fault opposes the dip of the seam 444

8.54 Given the angle 8 between the full dip of the seam and the true bearing of the fault, to find the bearing of the line of contact 446

8.55 To find the true bearing of a fault when the downthrow of the fault is in the same general direction as the dip of the seam 449

8.56 Given the angle 8 between the full dip of the seam and the true bearing of the fault, to find the bearing of the line of contact 449

8.6 To find the bearing and inclination of the line of intersection (AB) of two inclined planes 450

Exercises 8 (b) (Faults) 452

Exercises 8 (c) (General) 454

9 AREAS 457

9.1 Areas of regular figures 457

9.11 Areas bounded by straight lines 457

9. 12 Areas involving circular curves 459

9. 13 Areas involving non-circular curves 460

9.14 Surface areas 461

9.2 Areas of irregular figures 471

9.21 Equalisation of the boundary to give straight lines 471

9.22 The mean ordinate rule 472

9.23 The mid-ordinate rule 473

9.24 The trapezoidal rule 473

9.25 Simpson's rule 474

9.26 The planimeter 477

9.3 Plan areas 481

9.31 Units of area 481

9.32 Conversion of planimetric area in square inches into acres 482

9.33 Calculation of area from co-ordinates 482

9.34 Machine calculations with checks 488

10 VOLUMES 501

10. 1 Volumes of regular solids 501

10.2 Mineral quantities 509

Exercises 10 (a) (Regular solids) 511

10.3 Earthwork calculations 513

10.31 Calculation of volumes from cross-sectional areas 513

Exercises 10 (b) (Cross- sectional areas) 523

10.32 Alternative formulae for the calculation of volumes from the derived cross-sectional areas 525

10.33 Curvature correction 535

10.34 Derivation of the eccentricity e of the centroid G 537

10.4 Calculation of volumes from contour maps 543

10.5 Calculation of volumes from spot-heights 543

10.6 Mass-haul diagrams 544

10.61 Definitions 544

10.62 Construction of the mass-haul diagram 545

10.63 Characteristics of the mass-haul diagram 546

9.4 Subdivisions of areas 490

9.41 The subdivision of an area into specified parts from a point on the boundary 490

9.42 The subdivision of an area by a line of known bearing 491

10.64 Free-haul and overhaul 546

Exercises 10 (c) (Earthwork volumes) 552

9.43 The sub-division of an area by a line through a known point inside the figure 492

11 CIRCULAR CURVES 559

11.1 Definition 559

11.2 Through chainage 559

11.3 Length of curve L 560

11.4 Geometry of the curve 560

11.5 Special problems 561

11.51 To pass a curve tangential to three given straights 561

11.52 To pass a curve through three points 563

Exercises 11(a) 566

11.53 To pass a curve through a given point P 567

Exercises 11(b) (Curves passing through a given point) 571

11.54 Given a curve joining two tangents, to find the change required in the radius for an assumed change in the tangent length 572

11.6 Location of tangents and curve 575

11.7 Setting out of curves 576

11.71 By linear equipment only 576

11.72 By linear and angular equipment 580

11.73 By angular equipment only 580

Exercises 11(c) 588

11.8 Compound curves 591

Exercises 11(d) (Compound curves; 599

11.9 Reverse curves 600

Exercises 11(e) (Reverse curves) 605

Exercises 9 497

12 VERTICAL AND TRANSITION CURVES 607

12.1 Vertical curves 607

12.2 Properties of the simple parabola 608

12.3 Properties of the vertical curve 609

12.4 Sight distances 611

12.41 Sight distances for summits 611

12.42 Sight distances for valley curves 613

12.43 Sight distance related to the length of the beam of a vehicle's headlamp 615

12.5 Setting-out data 616

Exercises 12(a) 624

12.6 Transition curves 627

12.61 Superelevation 627

12.62 Cant 628

12.63 Minimum curvature for standard velocity 628

12.64 Length of transition 629

12.65 Radial acceleration 629

12.7 The ideal transition curve 630

12.8 The clothoid 632

12.81 To find Cartesian co-ordinates 632

12.82 The tangential angle 633

12.83 Amount of shift 633

12.9 The Bernouilli lemniscate 634

12.91 Setting out using the lemniscate 635

12. 10 The cubic parabola 636

12.11 The insertion of transition curves 637

12.12 Setting-out processes 640

12. 13 Transition curves applied to compound curves 644

Exercises 12(b) 649