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Geologic Geometry

Learn about the importance of topographic maps in solving problems in mining and civil engineering. Understand how to read and interpret contour lines to accurately represent the irregularities of the earth's surface. Explore profiles and vertical sections to visualize the terrain.

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Geologic Geometry

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  1. Geologic Geometry

  2. TOPOGRAPHIC MAPS

  3. Many problems in mining and civil engineeringare solved by graphic communication

  4. Mining and civil engineers make frequent use of topographic maps These are graphical representations of the irregularities of the earth’s surface in a single view

  5. TOPOGRAPHIC MAPS A topographic map is a two-dimensional representation of a portion of the three dimensional surface of the earth. It illustrates both natural and manmade features. • These features are divided into three groups: • Relief: including hills, valleys and mountains • Water: including lakes, ponds and rivers • Culture: including works of man such as roads and buildings

  6. If I walk along a 45m contour line, I am walking along a horizontal line 45m above the H.P. If I walk along a 50m contour line, I am walking along a horizontal line 50m above the H.P. Mining and civil engineers make frequent use of contour maps Contour lines are lines drawn on a map connecting points of equal elevation The contour interval measurement is the vertical distance between adjacent contour lines

  7. 0 0 CONTOUR LINES ON A CONE If a portion of the terrain were a right cone, the contour lines would be a series of circles 40 30 20 10 40 30 20 10

  8. horizontal datum plane 10m contour interval contour lines on map horizontal projection plane CONTOUR LINES ON THE EARTH’S SURFACE Unlike a cone, the surface of the earth is irregular; therefore the contour lines are irregular The earth’s surface is intersected by a series of horizontal cutting planes at 10 m intervals This results in contour lines in the plan view (ordnance map)

  9. INTERPOLATIONThe plotting/drawing of contour lines on a map

  10. The elevation of each intersection point on the grid is determined first 49.3 50.6 47.1 53.0 The contour lines are plotted next 46.2 51.2 54.1 48.5 47.3 45.5 49.9 52.2 43.8 47.3 49.5 45.2 PREPARATION OF CONTOUR MAPS To prepare a topographic map, a survey of the area is made to determine the elevation of an adequate number of strategically selected points One method for locating points is the grid system shown below

  11. Q P 49.3 50.6 49.3 50.6 47.1 53.0 B B R S 51.2 48.5 46.2 51.2 54.1 48.5 47.3 45.5 49.9 52.2 43.8 47.3 49.5 45.2 PREPARATION OF CONTOUR MAPS The elevation of each intersection point on the grid is shown. Assuming that the contour lines are to be plotted at 2 m intervals, locate the contour lines in grid B. The difference in elevation of points P and Q is 1.3 m We assume that the slope between the two points is constant Contour line 50 passes through a point seven-thirteenths of the distance between P and Q The difference in elevation of points R and S is 2.7 m Contour line 50 passes through a point five-ninths of the distance between points R and S

  12. 49.3 50.6 Let’s enlarge the grid for clarity 48.5 51.2 PREPARATION OF CONTOUR MAPS 13 7 B 50 50 5 9

  13. 13 7 2 49.3 50.6 An additional point on the contour line may be determined by interpolation along one diagonal of the grid 5 48.5 51.2 5 9 PREPARATION OF CONTOUR MAPS 50 50 50

  14. 23 16 26 32 The elevation of each intersection point on the grid is shown. Assuming that the contour lines are to be plotted at 10 m intervals, locate the contour lines in the grid.

  15. 7 4 16 23 4 7 10 9 26 32 2 3 The elevation of each intersection point on the grid is shown. Assuming that the contour lines are to be plotted at 10 m intervals, locate the contour lines in the grid. 20 20 30 30

  16. 16 23 26 32 Alternative method of solution using a Template The elevation of each intersection point on the grid is shown. Assuming that the contour lines are to be plotted at 10 m intervals, locate the contour lines in the grid. 20 20 30 30

  17. 20 1 2 30 3 30 40 1 50 2 The elevation of each intersection point on a triangular grid is shown. Assuming that the contour lines are to be plotted at 10 m intervals, locate the contour lines in the grid. 40

  18. PROFILESA profile is the line of intersection of the earth's surface and any vertical cutting plane.

  19. 30 20 10 10 20 30 VERTICAL SECTION OF A CONE HYPERBOLA When a vertical plane intersects an upright right cone the resulting section is a ………… B A Draw a vertical section (profile) on the line AB

  20. The accompanying drawing shows ground contours at ten-metre vertical intervals on a map. In the space provided draw a vertical section (profile) on the line DE. PROFILE

  21. Determine the profile along a line joining G to F The line of sight must pass the outline of the profile A wind turbine stands vertically on the ground at G. Determine the minimum height of the turbine so that it is visible from the ground at F. MINIMUM HEIGHT LINE OF SIGHT

  22. EARTHWORKS

  23. PROBLEM The drawing on the next slide shows ground contours at five- metre vertical intervals. PQ is the line of a proposed roadway having the following specifications. (i) formation width as given; (ii) formation level at P is 55m; (iii) P to Q is level; (iv) side slopes for cuttings 1 in 1; (v) side slopes for embankments 1 in 1.5. (a) Draw a Vertical section (profile) on the line PQ. (b) Show the earthworks necessary to accommodate the roadway.

  24. Fill needed here to achieve required level Cut needed here to achieve required level SOLUTION 80 75 70 65 60 55 50 45 40 35 30 ROAD LEVEL 30 35 35 40 45 55 50 60 65 70 75 30 35 40 75 45 70 65 50 60 SPACING FOR FILL = 7.5mm SPACING FOR CUT = 5mm 60 50 65 45 70 75 40 80 35 30 80

  25. PROBLEM The drawing on the next slide shows ground contours at five-metre vertical intervals. ABC is the line of a proposed roadway having the following specifications. (i) formation width as given; (ii) formation level at A is 65m; (iii) A to C is level; (iv) side slopes for cuttings 1 in 1; (v) side slopes for embankments 1 in 1.5. (vi) O is the centre of the circular part of the roadway. Show the earthworks necessary to accommodate the roadway.

  26. SOLUTION LOCATE ALL POINTS WHERE 65M CONTOURS PASS THROUGH ROAD. NO CUT/FILL REQUIRED HERE CUT FILL CUT FILL

  27. PROBLEM The drawing on the next slide shows ground contours at five-metre vertical intervals. AB is the centreline of a proposed level roadway. MNOP is the outline for a proposed level parking area. The roadway and picnic area have the following specifications: (i) Width of the roadway is as shown. (ii) Formation level is 65m (iii) Side slope for cuttings 1:1.5 (iv) Side slopes for embankments 1:1 On the drawing show the outline of the earthworks necessary to accommodate the roadway and parking area.

  28. SOLUTION EACH EDGE OF CAR-PARK IS AT 65m. FIND THE EARTHWORKS REQUIRED FOR EACH EDGE SEPERATELY RIDGE 65 VALLEY 70 75 80

  29. PROBLEM The drawing on the next slide shows ground contours at five-metre vertical intervals. AB is the centreline of a proposed roadway having the following specifications. (i) formation width as given; (ii) formation level at A is 140m; (iii) A to B is 1:10 rising; (iv) side slopes for cuttings 1 in 1.5; (v) side slopes for embankments 1 in 2. Show the earthworks necessary to accommodate the roadway.

  30. SOLUTION R=10m R=7.5m 1:10 RISING 145 145 160 140 165 150 135 170 155 175 130

  31. PROBLEM The drawing on the next slide shows ground contours at five-metre vertical intervals. ABC is the centreline of a proposed roadway having the following specifications. (i) formation width as given; (ii) formation level at A is 105m; (iii) A to B is level, B to C is 1:12 falling (iv) side slopes for cuttings 1 in 1; (v) side slopes for embankments 1 in 2. Show the earthworks necessary to accommodate the roadway.

  32. SOLUTION

  33. Mining

  34. TERMINOLOGY Stratum – A layer/plane of ore which has uniform thickness and is usually inclined to the earths crust. Headwall – The top surface of the stratum Footwall – The bottom surface of the stratum. Dip – The angle the stratum makes with the horizontal plane. Strike – The direction of a horizontal line on the surface of the stratum measured relative to North. Outcrop – The point at which a section of the stratum of ore lies at or above the surface of the ground

  35. The accompanying drawing shows ground contours at ten-metre vertical intervals on a map. A, B and C are outcrop points on a stratum of ore. Determine the dip and strike of the stratum. Draw the complete outline of the outcrop. 90 80 C 70 STRIKE OUTLINE OF OUTCROP 60 EDGE VIEW OF PLANE (STRATUM OF ORE) B 50 DIP A

  36. Determine the inclination of plane ABC to the Horizontal Plane EDGE VIEW OF PLANE ABC Ø STRIKE

  37. The slope between points A and B is assumed to be constant. Dividing the line into three equal parts will locate heights of 20m and 30m on the stratum. A, B and C are outcrop points on a stratum of ore whose altitudes are 10m, 40m and 20m respectively. Determine the dip and strike of the stratum without drawing the elevation. B 40m C 20m A 10m DIP C STRIKE B 20m A

  38. PROBLEM On a contour map A and B are two points whose altitudes are 90m and 100m respectively. On the map A is located 65m south of B. A skew borehole at A is drilled in a south-westerly direction in plan and has an actual inclination of 60° to the horizontal plane. It reveals the top and bottom surfaces of a stratum of ore at altitudes of 60m and 30m, respectively. A skew borehole at B is drilled in a south-easterly direction in plan and has an actual inclination of 40° to the horizontal plane. It reveals the top and bottom surfaces of the stratum at altitudes of 80m and 10m, respectively. Determine the dip, strike and thickness of the stratum.

  39. SOLUTION LINE ON HEADWALL OF STRATUM Bh Bh Ah LINE ON FOOTWALL OF STRATUM Ah Af Bf Af 60° 40° DIP Bf Bh STRIKE PARALLEL TO Af – Bf IN PLAN Ah Bf Af

  40. APPARENT DIP If a vertical section plane is taken at right angles to the strike of a stratum, then the dip is at its maximum, the true dip is found. When the angle between the cross section and the strike is anything less than 90° then the apparent dip is some value less than the true dip.

  41. On a contour map A and B are two points whose altitudes are 85m and 110m respectively. On the map B is located 70m north of A. A skew borehole at A is drilled in a south-westerly direction in plan and has an actual inclination of 55° to the horizontal plane. It reveals the top and bottom surfaces of a stratum of ore at altitudes of 60m and 40m, respectively. A skew borehole at B is drilled in a south-easterly direction in plan and has an actual inclination of 60° to the horizontal plane. It reveals the top and bottom surfaces of the stratum at altitudes of 90m and 10m, respectively. Determine the dip, strike and thickness of the stratum. Determine the apparent dip of the stratum on a vertical section through A that tends in a southerly direction. PROBLEM

  42. SOLUTION B @110m Bh Bh A @ 85m PARALLEL TO Ah-Bh IN ELEVATION Ah Ah P Af Af Q R S Bf Bf 60° 55° THICKNESS DIP B PARALLEL TO Ah-Bh IN PLAN Bh STRIKE APPARENT DIP Bf Q P A Ah Af R S SOUTH

  43. PROBLEM • On a contour map A and B are two points whose altitudes are 100m • and 110m respectively. On the map B is located 85m north-east of A. • A skew borehole at A is drilled in a north-easterly direction in plan and has an actual inclination of 60° to the horizontal plane. It reveals the top and bottom surfaces of a stratum of ore at distances of 35m and 100m respectively from A. • A skew borehole at B is drilled in a south-easterly direction in plan and has an actual inclination of 50° to the horizontal plane. It reveals the top and bottom surfaces of the stratum at distances of 45m and 95m respectively from B. • Determine the dip, strike and thickness of the stratum. • Another skew bore-hole at A is drilled in a south-westerly direction in plan and has a true inclination of 45° to the bore-hole already drilled at A. Determine the distance from A to the bottom surface of the stratum along this bore-hole and also find it's true inclination to the stratum.

  44. REQUIRED DISTANCE SOLUTION HORIZONTAL LINE B PARALLEL TO Af-Bf IN ELEVATION A P Bh A Ah Ah Q Ah Q Af 45° Bf Bh Af Af R R 50° 60° DIP Bf B XY LINE PARALLEL TO BOREHOLE No.3 Bh Af STRIKE Bf Ah P A PARALLEL TO Af-Bf IN PLAN BOREHOLE No.3

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