Structural Geology (Geol 4013/6013) Name: _________________ Lab Exercise 3 Questions (45 pts); Mohr Circle (15 pts); Glacier (15 pts); Three-point problem (25 pts). Strain ellipse, deformed fossils, pebbles, and glacier ice, Mohr circle Three-point problem Attached: Figure 12.32; Figure 12.33; Figure 12.34; Figure 12.35; Figure 12.37; Figure 12.38; Map 6. Study Chapter 4 (Section 4.6, Page 80) and Chapter 12 of the Lab Manual by Donal M. Ragan: “Structural Geology, an Introduction to Geometrical Techniques”. Chapter 12. Do the following problems from the “Exercises 12.12” on page 343 of the Lab Manual: Problem 1 (Page 343). Topic: Strain Ellipse Using the circle and ellipse of Figure 12.32 (attached!), graphically determine the stretch S and the angle of shear () associated with a radius making an angle of ’= +30o with the major axis (in deformed state). Check your result with a Mohr circle. Hint: The maximum principal stretch is S1 = a/r, and the minimum principal stretch is S2 = b/r, where a and b are the long and short semi-axis of the strain ellipse, respectively, and r is the radius of the initial circle. In general, S=l’/l, or in this case, S=r’/r, where r’ is the length of any semi-axis of the strain ellipse at any angle (’) to the major principal axis. Notice that in a circle, any tangent to the circle is perpendicular to all radii. For an ellipse, such as the strain ellipse, however, the tangent is not perpendicular to the ‘radii’ of the ellipse except across the principal directions. The deviation from perpendicular gives the angle (see my lecture ppt on deformation). To solve this problem, draw r’ at the ’=30 angle (ccw) from the long axis of the strain ellipse. Define point P where r’ touches the ellipse. Draw a tangent to the ellipse at P. Notice that r’ is not perpendicular to the tangent to the ellipse (it used to be perpendicular to the tangent to the circle before it was deformed!). To find out how much it has rotated, draw a perpendicular to the tangent to the ellipse through the center of the ellipse. Find the angle between this line and r’ (this is the angle). Note: Measure the lengths to the nearest millimeter and the angles to the nearest degree! Questions 1. What is the stretch (S), along r’? a. 0.08 b. 1.166 c. 0.58 d. 1.08 e. -0.86 2. What is the angle of shear ()? a. 40 b. -84 c. 68 d. 84 e. 12 Structural Geology (Geol 4013/6013) Lab 3. Page 1 Chapter 12, Problem 2 (Page 343). Topic: Deformed pebbles (see Section 12.4, Page 306 to learn the method). Given the 9 deformed pebbles (i=1-9) in Figure 12.33 (attached!), determine the final shape ratio (Rf) for each pebble (Note: R = (long axis)/(short axis), and the orientation of its long axis of each pebble (’i) is with respect to the lower, long edge of Figure 12.33 (this is the reference line!). 3. What is the final ratio (Rf) for pebble 1? a. 1.9 b. 1.5 c. 5.2 d. 2.1 e. 3.1 4. What is the final ratio (Rf) for pebble 2? a. 4.3 b. 1.6 c. 3.8 d. 2.6 e. 1.1 5. What is the final ratio (Rf) for pebble 3? a. 2.4 b. 1.2 c. 3.2 d. 4.4 e. 5.9 6. What is the final ratio (Rf) for pebble 4? a. 1.4 b. 3.9 c. 2.4 d. 4.6 e. 3.3 7. What is the final ratio (Rf) for pebble 5? a. 4.8 b. 1.5 c. 3.1 d. 2.0 e. 1.1 8. What is the final ratio (Rf) for pebble 6? a. 1.9 b. 2.8 c. 3.8 d. 1.2 e. 4.2 9. What is the final ratio (Rf) for pebble 7? a. 1.7 b. 4.2 c. 3.0 d. 2.9 e. 3.6 10. What is the final ratio (Rf) for pebble 8? a. 1.0 b. 2.2 c. 3.8 d. 2.6 e. 1.9 11. What is the final ratio (Rf) for pebble 9? a. 1.0 b. 1.5 c. 3.2 d. 2.8 e. 3.8 12. What is the angle (’), measured from the long edge (reference line), for pebble 1? a. 46 b. 64 c. 26 d. 86 e. 20 13. What is angle (’) from the long edge (reference line) for pebble 2? a. 60 b. 32 c. 56 d. 45 e. 80 14. What is angle (’) from the long edge (reference line) for pebble 3? a. 102 b. 60 c. 22 d. 48 e. 72 15. What is angle (’) from the long edge (reference line) for pebble 4? a. 82 b. 37 c. 48 d. 28 e. 18 16. What is angle (’) from the long edge (reference line) for pebble 5? a. 42 b. 15 c. 90 d. 20 e. 68 17. What is angle (’) from the long edge (reference line) for pebble 6? a. 97 b. 70 c. 38 d. 14 e. 48 18. What is angle (’) from the long edge (reference line) for pebble 7? a. 16 b. 46 c. 50 d. 74 e. 98 19. What is angle (’) from the long edge (reference line) for pebble 8? a. 76 b. 12 c. 38 d. 26 e. 104 20. What is (’) angle from the long edge (reference line) for pebble 9? a. 28 b. 44 c. 101 d. 81 e. 66 Plot the Rf vs. ’ on a graph paper (see Fig. 12.5b as a guide). This is a case of Rs > Ri (see Fig. 12.4) 21. What is the Rmax from the plot (i.e., the top of the ‘drop-shaped’ curve!)? a. 1.6 b. 2.4 c. 4.9 d. 3.2 e. 1.8 22. What is the Rmin from the plot (i.e., the bottom of the ‘drop-shaped’ curve!)? a. 3.0 b. 1.2 c. 2.6 d. 4.4 e. 3.8 23. What is the strain ratio, Rs = ( Rmax Rmin) (see eqn. 12.2, page 307) a. 3.0 b. 2.0 c. 3.8 d. 4.8 e. 1.0 24. What is the initial grain ratio, Ri = (Rmax /Rmin) (see eqn. 12.2, page 307) a. 2.9 b. 4.2 c. 3.4 d. 1.6 e. 3. 8 Structural Geology (Geol 4013/6013) Lab 3. Page 2 Chapter 12, Problem 3 (Page 344). Using the group of 8 deformed brachiopods of Figure 12.34 (attached!) apply the Wellman method to determine the strain ratio (Rs). Procedure for Wellman method: Draw an arbitrary line AB (of any length, say 5 cm) parallel to the lower, long edge of Figure 12.34 on a transparent paper. While this line is kept parallel to the lower edge, put point A at the intersection of the median and hinge lines of brachiopod #1. Hold the tracing paper in place without moving or turning it. With a straight edge ruler, trace the median (m’) and hinge line (h’) of the deformed brachiopod #1 through point A on the tracing paper. Then put point B on the same brachiopod #1, and again, while AB is kept parallel to the lower edge, draw the two lines (m’ and h’) through point B (make sure the paper does not turn or move while you do this). The four lines make a parallelogram, with two intersection points. Mark these two points with small ‘x’ (make these fine points bold so that you can see them well). Erase the lines! Repeat the above steps for all the remaining 7 fossils. Make sure that AB remains parallel to the long, lower (or upper) edge of the figure at all times. The extensions of the two lines through points A and B, for each fossil, intersect at two points. Mark all of these with bold ‘x’ points. By now, you must have 16 points (x) for the 8 brachiopods (erase all lines). All of these points lie on a rough ellipse which approximates the strain ellipse. Read ellipse’s long and short semi-axes, and then find the strain ratio Rs by dividing the long semi-axis over the short semi-axis. 25. What is the strain ellipse ratio, Rs? a. 2.42 b. 2.20 c. 3.8 d. 3.2 e. 1.53 26. What is the angle between the long axis of the strain ellipse and line AB? (clockwise or counter clockwise from point A) a. 32 CCW b. 64 CW c. 26 CW d. 16 CCW e. 43 CCW Structural Geology (Geol 4013/6013) Lab 3. Page 3 ____________________________________________________________________________________ /15 pt Chapter 12, Problem 4 (Page 344). Using the Mohr circle for strain (see Page 318), determine the orientation and shape of the strain ellipse from the two deformed brachiopods of Figure 12.35 (attached!). Procedure for Mohr Circle: Label the brachiopod (in the physical, real world) on the left side of Figure 12.35 as ‘A’, and the brachiopod on the right as ‘B’. Notice that the initially perpendicular hinge (h) and median (m) lines of the two brachiopods are not perpendicular anymore! Measure the amount and sense of the angle of shear () for brachiopod ‘A’ (A) and brachiopod ‘B’ (B) in the physical plane (i.e., real world of Figure 12.35). To do this, you need to draw a line perpendicular to the hingelines of each brachiopod, and then measure the angle between this line and the median line. Measure the angular shear (A and B) for each fossil; note the sense for each (cw or ccw). Remember that the sense of rotation (ccw or cw) is the same in both the physical plane (i.e., the image of the two fossils) and in the Mohr circle. Extend the hinge line (i.e., long line) of both brachiopods until they intersect (measure the angle as ’). 27. What is the angle (’) between the two extended hinge lines? a. 26 b. 63 c. 18 d. 46 e. 88 28. What is the angular shear for fossil A (A)? a. 8 CCW b. 35 CW c. 42 CW d. 19 CCW e. 39 CW 29. What is the angular shear for fossil B (B)? a. 38 CW b. 22 CW c. 12 CCW d. 31 CCW e. 28 CW Plot an arbitrary circle of any size (e.g., radius 3 cm) on a transparent tracing paper. Draw two radii (label them as A and B) through the center of this circle, with an angle of 2’ between them. Draw the coordinate axes of the Mohr Circle ( ’ vs. ’) on a mm-scale graph paper. On this graph paper, draw two lines radiating from the origin, each inclined at the angle to the horizontal axis. Note: CCW is up, CW is down (it is same in the physical and Mohr circle worlds!). Label these lines as A or B. Overlay the tracing paper on the graph paper, while holding the center of the circle on the x-axis. Rotate the circle (i.e., the tracing paper) until each of its radii (A and B) intersects its corresponding line that emanates from the origin (make sure that the center of the circle is on the x axis at all times!) Determine the two points at which the circle intersects the x-axis. Mark these two points as k’1 and k’3. Read the angle (2‘) between the line connecting the k1 and any of the arms (i.e., the hingeline of fossil A or B). Half of this angle (i.e., ‘) gives the orientation between the maximum principal stretch (S1) relative to the hingeline in the real world. Read the ratio k’1/k’3 and determine the strain ratio (Rs) (Note: k cancels out) 30. What is the strain ratio? a. 1.0 b. 2.8 c. 1.5 d. 3 e. 3.4 31. What is the angle between the hingeline of brachiopod ‘B’ and the maximum principal stretch, S1? a. 13o b. 24o c. 42o d. 32o e. 43o Structural Geology (Geol 4013/6013) Lab 3. Page 4 Chapter 12, Problem 6 (Page 345, Figure 12.37). Determine the stretch (S) of the boudinaged object shown in Figure 12.37 (attached!), using the data in Table 12.1 (Page 345) for each of its segments, applying the method of Ramsay, Hossain, and Ferguson. See Page 322 (Fig. 2.16) for guide! 32. What is the original length (lo) of the boudinaged object? a. 82 mm b. 40 mm c. 28 mm d. 66 mm e. 33 mm 33. What is the final length (l’) of the boudin? a. 140 mm b. 79 mm c. 100 mm d. 80 mm e. 82 mm 34. What is the elongation (e) of the boudin? a. 0.78 b. 0.51 c. 0.97 d. 1.12 e. 1.48 35. What is the stretch (S) of the boudin using Ramsay’s method? a. 1.0 b. 1.51 c. 1.97 d. 2.12 e. 2.48 36. What is the stretch (S) of the boudin using Hossain’s method? a. 1.0 b. 1.18 c. 1.30 d. 2.23 e. 1.64 /15 pts Chapter 12, Problem 7 (Page 345). Three stakes were placed on the surface of a glacier to form an equilateral triangle 10 m on each side (undeformed triangle on Fig. 12.38a). Figure 12.38 is attached! After one year, the positions of the stakes (i.e., A, B, C) were resurveyed with differential GPS (Fig. 12.38b). Notice that points A, B, C are displaced to points A’, B’, and C’. Determine the strain that accumulated over this time span in the glacier. NOTE: Follow the procedure for the problem on page 325 and 326 to solve this problem. Procedure: Label the undeformed ABC triangle’s side CB as a, AC as b, and AB as c. After one year, these undeformed sides a, b, and c, become deformed into a’, b’, and c’ (in triangle A’B’C’). Measure the length of all these sides (in mm) in both the undeformed and deformed states (a, b, c, a’, b’, c’). Notice that the angle between these three sides are also changed (initial and final angles are given on Figure 12.38). Calculate stretch (S = l’/lo) for sides a, b, and c (i.e., Sa, Sb, Sc). Calculate the principal reciprocal quadratic elongation for the three sides (i.e., ’a, ’b, ’c), where =S2, and ’=1/, or ’ =1/ S2. 37. What is the stretch (S) for side CB (a)? a. 2 b. 3 c. 2.4 d. 1.0 e. 3.4 38. What is the stretch for side AC (b)? a. 1.17 b. 2.66 c. 1.88 d. 1.42 e. 2.20 39. What is the stretch for side AB (c)? a. 1.22 b. 0.84 c. 0.40 d. 0.30 e. 1. 43 40. What is the reciprocal quadratic elongation for side CB (a), ’a? a. 3.2 b. 3.9 c. 2.4 d. 1.8 e. 1.0 41. What is the reciprocal quadratic elongation for side AC (b), ’b? a. 1.17 b. 2.66 c. 1.80 d. 0.73 e. 0.20 42. What is the reciprocal quadratic elongation for side AB (c), ’c? a. 2.22 b. 1.98 c. 1.42 d. 1.10 e. 3.21 43. What is the strain ratio (Rs)? a. 2.15 b. 1.50 c. 2.5 d. 2.8 e. 1.10 Structural Geology (Geol 4013/6013) Lab 3. Page 5 /25 pts Topic: Three-point problem and completion of map Map 6 (Bennison, attached!) is a topographic map that partially exposes a coal seam at points A, B, and C. The elevations of these points are readable from the contour lines! Use the three-point problem procedure described in Section 4.6 (Page 80) of the Lab Manual. Procedure: Points A, B, and C have different elevations. We first need to draw the structural contours (the strike of the coal seam). Drawing the structural contours along the coal seam requires that we find two points of equal elevation. To find these two points, connect the lowest point (A at 200 m) to the highest point (C at 600 m) with color (red) pencil. Using your ruler, graduate the distance between points A and C into 4 segments. In this case, since the elevation at C=600 and elevation at A=200 meters, then each of the four segment represents 100 meters. Somewhere along the line AC, there is a point that has the same elevation as point B (400 m). We want to find it. This point (B’) lies 2 units (200 m) above point A, along line AC. Thus, both B and B’ have the same elevation of 400 m. Connect the lowest point A (200 m) to the intermediate point B (400 m) with a straight line with a color (blue) pencil. Connect B and B’ (both have elevation of 400 m) with a color pencil (green). This is the strike of the coal seam because it is horizontal (remains at the same elevation along its length). Draw other green lines, parallel to the strike, through points C and A. By now, you have three parallel structural contours at elevation 600 m (point C), elevation 400m (point B), and elevation 200 m (point A); all drawn in green. These are the strike of the coal seam. Now, you can determine the dip. Draw a line from point A (which is at 200 m) perpendicular to the structural contour 400 m. This is the horizontal distance (x) between two structural contours (strikes) whose difference in the elevation (h) is 200 m (400-200). Use the scale of the map to determine the length of x in meters. Use trigonometry and find the true dip () of the coal seam (tan = h /x) Draw all other structural contours (with green pencil) at steps of 100 meters. Label all the structural contours (green lines) with elevation values. Find the intersection of the green structural contours (strikes) with the topographic contours of the same elevation. Put a black dot () at each intersection point. Connect all these dots to find the exposure of the coal seam (draw this line in black) Assume that the lower seam, which is 200 m below the upper seam, has the same attitude (i.e., it has the same strike and dip as the upper seam). Re-label all the structural contours with a different color pencil 200 m lower for the lower contact. Notice that a structural contour that has an elevation for the upper seam of say 400 m, becomes 200 m for the lower seam. Find the structural contour that goes through point D. If none goes through, find one by extrapolation, i.e., finding and drawing the 100 m, 75m, 50m, and 25m structural contours. Here is an example for finding how much we have to drill at point D to reach the lower contact. If the red structural contour 700 m goes through point D, and knowing that point D has a surface elevation of 400 m, then we have to drill 300 m to reach the lower seam. Do the following: 44. What is the strike and dip of the coal seam? a. N18E, 32NW b. N51E, 11SE c. N18E, 32SE d. 198, 40SE e. N46E, 12NW 45. At what depth would the seam be encountered in a borehole sunk at point D? a. 0 m b. 100 m c. 200 m d. 300 m e. 400 m Would a coal seam 200 m below this coal seam also outcrop within the area of the map? If yes, draw the exposure of this coal seam on the map with a different color pencil! Structural Geology (Geol 4013/6013) Lab 3. Page 6

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