# STRUCTURAL GEOLOGY EXERCISES with Glaciotectonic Examples – Part III

## 7. EQUAL-AREA STEREONET I

### Schmidt stereonet

The equal-area or Schmidt stereonet is a different stereographic projection on which size is preserved, but angles and shape are distorted—see Fig. 7-1. Each quadrangle on the stereonet is equal in size or area, whereas shape of quadrangles varies from almost square near the center to narrow, curved rectangles near the edge. Unlike the Wulff stereonet, on which meridians and parallels are circular arcs, the meridians and parallels of the Schmidt stereonet are oval curves.

Plotting of lines and planes on the Schmidt stereonet is carried out following exactly the same procedures as used on the Wulff stereonet. All the operations described in the previous exercise may be accomplished just as easily on either stereographic projection. Because it maintains size, the Schmidt stereonet is visually more pleasing than the equal-angular stereonet, and the Schmidt stereonet has proven more popular.

The Schmidt stereonet has one important advantage--it preserves area. Therefore, the density of plotted data may be analyzed. In many situations, dozens or even hundreds of measurements of linear or planar features may be made, and all the data could be plotted on a single Schmidt stereonet to display the over all structural pattern.

### Petrofabric analysis

Petrofabric refers to the size, shape, and arrangement of grains which make up a body of rock or sediment. Size and shape of grains vary from flat clay flakes <2 microns long, to intergrown crystals a few mm or cm long, to large rounded boulders >1 m in diameter. The grains may be arranged in a linear fabric, called lineation, or in a planar fabric, called foliation. The terms lineation and foliation are commonly applied to metamorphic rocks. However, these terms may be used for any type of rock or sediment whose fabric was primarily created by deformation. The orientation of lineation is given by trend and plunge; foliation is specified by strike and dip.

Petrofabric is normally related to the larger structures formed at the same time the fabric developed. For example, lineation may parallel fold axes, or foliation may parallel fold axial planes. The petrofabric of a rock reflects the internal changes or strain that occurred within the rock body during deformation (further discussion in exer. 9). Thus, petrofabric analysis could give important additional information for overall structural interpretation.

### Till fabric

Consider for example till fabric, a kind of lineation. The orientations of long axes of elongated pebbles embedded within the till are measured to ascertain the direction of ice movement. Typically 50 pebbles are measured, and all 50 axes plotted as points on a stereonet. The data points may then be contoured to show the pattern of data density. Another approach is to show each pebble axis as a tiny circle on the stereonet, in an attempt to retain the integrity of individual measurements, while giving the impression of a contouring effect.

The most common till fabric is parallel to ice movement with a majority of pebbles plunging at a slight angle in the upice direction—see Fig. 7-2. Fewer pebbles plunge slightly down ice, and still fewer pebbles may lie approximately transverse to ice flow. The transverse till fabric is rarer, but still appears often enough to warrant discussion. In this type, pebble axes are about equally divided, lying near-horizontal or plunging slightly left or right, perpendicular to ice flow. Mixed or ambiguous till fabrics also occur, but are of limited use for establishing ice movement direction.

A well-developed till fabric is thought to reflect the shearing action at the base of a glacier when the till was deposited. Elongated pebbles became aligned parallel either with the strike or dip of inconspicuous shear planes cutting through the till. The fabric was, thus, created by subglacial deformation penecontemporaneous with deposition of the till. Subsequent reworking of the till or a shift in ice-movement direction could alter or destroy the original fabric, however.

### Problem

The glaciation of northeastern Kansas was the greatest ice coverage to ever take place on the plains of central North America. The maximum ice limit reached west of the Missouri River, extending across northeastern Kansas and eastern Nebraska, far south of the younger Wisconsin glacial limit—see Fig. 7-3. The glaciation of Kansas is marked by the Independence Formation, which includes till and stratified drift. Age of the Independence Formation is in the range 600,000 to 720,000 years ago (Aber 1991), making it one of the oldest continental glaciations still preserved in the Pleistocene record.

The Wisconsin glaciation of the northern Great Plains consisted of two prominent ice lobes—Des Moines and James River, whose movements were controlled by bedrock topography. The lobes advanced southward following broad troughs either side of the Coteau des Prairies upland. The southern end of the Coteau des Prairies is underlain by a high bedrock ridge of resistant Sioux Quartzite, located in southwestern Minnesota and South Dakota.

The Independence glaciation likewise consisted of two ice lobes—Minnesota to the east and Dakota to the west. These two lobes were confluent over the crest of the Coteau des Prairies, but maintained their separate identities southward to the ice margin. The Minnesota lobe advanced southward through Iowa, into Missouri, and entered Kansas from the northeast. Conversely, the Dakota lobe came from the Dakotas, across Nebraska, and moved into Kansas from the northwest. Evidences from glaciotectonic structures and glacial striations confirm these two directions of ice advance in northeastern Kansas (Dellwig and Baldwin 1965).

Deposits of both ice lobes are present at the Independence Formation stratotype, near Atchison, Kansas—see Fig. 7-4. The lower till is a gray, wood-bearing, compact till that exhibits a large diapir and thrust faults. The diapir and thrusts extend up into fine sand and silt beds that were deposited in an ice marginal lake. Recumbent, isoclinal folds near the top of this sand were created by the ice advance that laid down the overlying upper till.

 Overview of Independence Fm. stratotype near Atchison, Kansas. Upper brown till (upper right) overlies deformed sand. Lower gray till (bottom left) is deformed in a diapir that intrudes sand in the middle of the section (behind ladder). Photo © J.S. Aber. Closeup view of the large diapir of lower gray till. Ibrahim Abdelsaheb (on ladder) is working at the head of the diapir. Photo © J.S. Aber.

The upper till was deposited by northwesterly ice movement. Axes of subjacent isoclinal folds plunge slightly to the northeast, and striations on bevelled boulder tops trend NW-SE. Till fabric measurements at site 4 (upper right, fig. 7-4) are displayed in two formats in Figure 7-5.

The rose diagram shows measurements grouped into 10-degree intervals according to plunge direction, but the plunge angles cannot be indicated. Pebble axes are plotted on the stereonet according to pebble shape: (1) large dots = blade and roller shapes, (2) small dots = disk and spheroid shapes. However, differences in pebble shapes do not appear to affect the orientations of pebble long axes. The fabric is clearly the parallel type: most pebbles plunge slightly north or northwest, a smaller group plunges gently in the opposite direction, and a few pebbles lie in transverse positions with steeper plunge angles. Ice movement from the NNW is indicated and agrees with other directional features.

Tables 7-1 and 7-2 (below) present fabric data for sites 1 and 3 from the lower gray till in the Independence Fm. stratotype (see fig. 7-4 for site locations). Forty-two measurements are given for site 1 and 50 for site 3. Use these data for the following.

1. Plot all 50 measurements for site 3 on a Schmidt stereonet.

Note for Stereowin: Enter trend/plunge values for a linear (+L) dataset. Save your data file periodically as you add more values.

2. Is the till fabric at site 3 parallel, transverse, or mixed in style?

Note for Stereowin: You should see 50 dots, each representing one pebble axis. To better visualized the pattern of data points: Plot, Contour, 1% Area.

3. What general direction of ice movement is indicated by the till fabric at site 3?

4. Plot all measurements for site 1 on a second Schmidt stereonet.

Note for Stereowin: Be sure to open a new linear dataset; do not add site 1 data to the site 3 dataset.

5. Is the till fabric at site 1 parallel, transverse, or mixed in style?

6. Is the site 1 till fabric similar to the till fabric at site 3? Describe any differences between the two fabrics.

7. Thrust faults that cut the diapir and adjacent sand near site 1 have an average strike/dip = 330/40°. Plot this fault as a plane on the stereonet for site 1. Describe any relationship between the fabric pattern and fault position that you see.

Note for Stereowin: Add a planar (+P) dataset for the thrust fault to the plot for till fabric at site 1.

8. Give an explanation for how the till fabric at site 1 was created.

9. What direction of ice movement was responsible for depositing and deforming the lower till?

10. Based on all the available information, determine which of the ice lobes deposited each of the tills at the Independence Formation stratotype.

Note: Turn in your stereonet plots (paper or pdf file) with written answers.

Table 7-1. Till fabric measurements from Independence Formation stratotype—site 1.
Trend/Plunge Trend/Plunge Trend/Plunge
1. 250/40 15. 080/30 29. 157/27
2. 115/09 16. 295/32 30. 072/33
3. 077/00 17. 322/74 31. 220/06
4. 340/56 18. 295/15 32. 288/71
5. 335/57 19. 328/78 33. 130/46
6. 063/04 20. 000/90 34. 035/08
7. 317/36 21. 040/36 35. 270/25
8. 136/49 22. 298/21 36. 003/55
9. 015/60 23. 280/26 37. 336/24
10. 072/35 24. 042/70 38. 090/62
11. 034/61 25. 342/54 39. 257/26
12. 088/39 26. 048/25 40. 044/65
13. 135/07 27. 255/15 41. 078/34
14. 338/71 28. 070/56 42. 110/10

Table 7-2. Till fabric measurements from Independence Formation stratotype—site 3.
Trend/Plunge Trend/Plunge Trend/Plunge
1. 153/06 18. 071/30 35. 068/15
2. 153/12 19. 133/02 36. 082/26
3. 044/03 20. 112/23 37. 315/17
4. 130/08 21. 054/21 38. 053/15
5. 244/20 22. 074/27 39. 160/42
6. 085/35 23. 008/40 40. 136/19
7. 062/36 24. 086/01 41. 341/21
8. 138/20 25. 038/32 42. 150/24
9. 250/08 26. 009/47 43. 247/18
10. 272/00 27. 105/10 44. 250/11
11. 070/19 28. 145/25 45. 110/00
12. 036/20 29. 245/24 46. 070/04
13. 104/35 30. 082/25 47. 140/13
14. 186/08 31. 230/27 48. 028/25
15. 170/23 32. 270/15 49. 108/00
16. 298/30 33. 165/30 50. 080/14
17. 160/06 34. 110/32

Based on Aber (1991).

### References

• Aber, J.S. 1985. Definition and model for Kansan glaciation. Ter-Qua Symposium Series 1:53-60.

• Aber, J.S. 1991. The glaciation of northeastern Kansas. Boreas 20:297-314.

• Dellwig, L.F. and Baldwin, A.D. 1965. Ice-push deformation in northeastern Kansas. Kansas Geological Survey, Bulletin 175, part 2.

• Nielsen, P.E. and Houmark-Nielsen, M. 1983. Till fabric. in, Ehlers, J. (ed.), Glacial Deposits in North-west Europe, p. 207-209. A. A. Balkema, Rotterdam.

## 8. EQUAL-AREA STEREONET II

### Plotting poles to planes

Plotting 50 pebble axes on the stereonet, as with till fabric diagrams, is straightforward. Each axis is a linear feature, and so, plots as a single point. In other situations, though, it may be necessary to deal with a large number of planar features. However, plotting 50 planes as arcs on the stereonet would create visual clutter and confusion.

To simplify plotting planes on the stereonet, it is convenient to plot the pole to each plane. The pole is a line running through the center of the projection sphere and perpendicular to the plane—see Fig. 8-1. The pole forms a 90° angle with the strike line and a 90° angle with the dip line. Thus, the pole is always be found in the opposite quadrant of the stereonet from the dip of the plane.

In the example (fig. 8-1), the plane strikes N50E and dips 45°SE. The plane's pole trends N40W (90° from strike) and plunges 45°NW. Note that plane dip angle plus pole plunge angle must equal 90°. The pole for a horizontal plane plots at the center of the stereonet, and the pole for a vertical plane plots as two "half points" on opposite edges of the stereonet. The procedure for plotting a pole to a plane is as follows.

1. Mark the plane's strike and dip compass directions on the outer edge of the stereonet.

2. Rotate the tracing so the strike mark points north and the dip mark falls on the E-W axis of the stereonet.

3. Count the dip angle outward from the center of the stereonet along the E-W axis toward the side opposite the dip, and mark that point.

4. Rotate the tracing back to its original position. The marked point represents the pole to the plane.

With a little practice, you will soon find it is easier to plot a plane's pole than to plot the actual plane. By plotting poles, a large number of planar measurements may be placed on a Schmidt stereonet in order to display and analyze various structural patterns.

### Constructing fold axes

The technique of plotting poles to bedding planes is particularly useful for determining the orientation of a fold axis, where the fold axis is not directly discernible. Such a plot is called a Pi diagram, and is analyzed on the assumption that the folds are parallel or cylindrical in style (Billings 1972).

Figure 8-2 illustrates a surface folded in a cylindrical style around three parallel fold axes. Poles perpendicular to the folded surface are indicated in various positions. Note that all of the poles are perpendicular to the fold axes, regardless of their positions on the folded surface. In fact, the poles define a plane which is perpendicular to the fold axes.

When plotted on a Schmidt stereonet, the poles to a cylindrical fold fall on an arc which represents the plane perpendicular to the fold axis. This arc is found simply by rotating the tracing until the poles line up along the same N-S meridian of the stereonet. The arc may then be sketched onto the tracing. As this arc is perpendicular to the fold axis, the pole to the arc represents the so-called "constructed fold axis" (fig. 8-2).

### Problem

Ice-shoved terrain and structures are well developed in the Limfjord region of northern Denmark—see Figs. 8-3, 8-4. The Limfjord is a large, relatively shallow estuary around whose margins ice pushing has folded and thrust Eocene strata into prominent ridges that stand >50 m above the surrounding lowlands (Gry 1940, 1979). The disturbed Eocene strata, which are part of the Fur Formation, consist of interbedded clayey diatomite, volcanic ash, and limestone layers. This formation was especially suspectible to glaciotectonism and is deformed in all surface exposures of the region (Aber and Ber 2007).

 Large exposure of Fur Formation in Harhøj gravel pit on the island of Mors, Limfjord estuary, northwestern Denmark. Overturned syncline on left leads into a sharply folded anticline on right, and the structure was truncated by overriding ice. Photo © J.S. Aber. Closeup view of overturned syncline. Fur Formation diatomite is wrapped around a core of glacial gravel (upper right). Notice the kink folds on the flanks of the main fold. Photo © J.S. Aber. Closeup view of chevron-style anticline (left) that continues into an overturned syncline with a core of glacial gravel (upper right). Photo © J.S. Aber.

The ice shoving presumably took place during the Weichselian glaciation, some 13,000 to 25,000 BP (Berthelsen 1978). All manner of glaciotectonic structures from beautifully concentric folding to overturned, thrust and contorted folding is developed in the Limfjord district. Northern Jylland was subjected to Weichselian ice advances from two directions: a) Norwegian advance directly from the north coming across the Skagerrak, and b) multiple northeasterly advances moving from Sweden across the Kattegat. Each glaciation formed ice tongues moving along topographic troughs that are now parts of the Limfjord.

 Fegge, a peninsula at the northern end of the island of Mors in the Limfjord. Aerial view (left) and ground shot (right) showing the cliff exposure on the side of the flat-topped hill. The cliff reveals folded and thrust bodies of Fur Formation. Photos © JSA and SWA.

At Ertebølle Hoved, ice pushing created parallel folds in the Fur Formation. Table 8-1 (below) lists 14 strike-and-dip measurements from this locality. On the basis of these data, use the Schmidt stereonet to:

1. Construct a Pi diagram by plotting each strike-and-dip measurement as a pole. Number each data point.

 Begin by entering the 14 strike-dip values for a planar dataset. The planes plot as arcs on the stereonet. Notice two sets of planes representing the two fold limbs. You may visually approximate the fold axis at the point where most of the planes intersect (red asterisk). Next convert the strike/dip values in Table 8-1 into pole values. Strike - 90 = pole trend; 90 - dip = pole plunge. For example, site 1: 279 - 90 = 189 (trend) and 90 - 44 = 46 (plunge). Enter the pole values in a new linear dataset. You should get a plot with 14 points that represent poles to planes.

2. Find the plane (arc) along which these poles approximately lie. Sketch in this arc; it is normal to the fold axis.

Note for Stereowin: For the display of poles, select Plot, Cylindrical Best Fit function.

3. Determine the trend and plunge of the pole to the plane sketched in step 2; this pole is the "constructed fold axis."

Note for Stereowin: In the previous step, axis #3 is the constructed fold axis. Its trend and plunge are given in the table of values (lower left).

4. Assuming that ice movement was more-or-less perpendicular to the fold axis, what direction of ice advance is indicated by the structural data at this locality?

5. Which Weichselian ice advance do you think deformed the Fur Formation at Ertebølle Hoved?

Table 8-1. Measurements of bedding planes from several places at Ertebølle Hoved, Denmark.
Site Strike/Dip Site Strike/Dip
1
279/44
8
194/13
2
275/54
9
194/12
3
273/41
10
175/26
4
256/24
11
171/24
5
223/26
12
144/37
6
214/13
13
141/26
7
197/13
14
139/24

Strike/dip according to the right-hand rule.
Adapted from Gry (1940, p. 588).

Note for Stereowin: As a final task, make a combined display of both planar (limbs) and linear (poles) datasets including the best-fit great circle. Turn in the pdf (or paper) display along with your written answers.

### References

• Aber, J.S. and Ber, A. 2007. Glaciotectonism. Developments in Quaternary Science 6, Elsevier, Amsterdam, 246 p.

• Berthelsen, A. 1978. The methodology of kineto-stratigraphy as applied to glacial geology. Geological Society of Denmark, Bulletin 27, Special Issue, p. 25-38.

• Billings, M.P. 1972. Structural Geology. Prentice-Hall, Englewood Cliffs, New Jersey, 606 p.

• Gry, H. 1940. De istektoniske forhold i moleromraadet. Meddr. Dansk Geologisk Forening 9:586-627.

• Gry, H. 1979. Beskrivelse til geologisk kort over Danmark, Kort bladet Løgstær, Kvartære aflejringer. Danmarks Geologisk Undersøgelse, I række, Nr. 26.

## 9. ROCK STRENGTH

### Stress and strain

Geologic bodies may be deformed by application of stress or pressure, and the resulting change in size or shape of the body is called strain. Stress is simply force per unit area, and may be expressed in such common units as: pounds per square inch (psi), kg/cm², or atmospheres (1 atm. = 14.7 psi or about 1 kg/cm²) pressure.

A stress unit commonly used in structural geology is the kilobar (kb), which is 1000 bars or approximately 1000 atmospheres. For continental rocks of medium density, a pressure of 1 kb is achieved at a depth of 3.5 to 4.0 km, and pressures of 10 kb is reached at the base of continental crust. It was under such high-pressure conditions that plutonic and metamorphic rocks now exposed in mountain belts and shields were created.

The high pressure imposed on deeply buried rocks due to weight of the overburden is uniform and equal in all directions. A similar, uniform pressure is experienced by underwater divers. Such stress in crustal rocks is called lithostatic pressure, and it is determined solely by the thickness and density of the over burden. It is compressive or positive stress, in contrast to tension which is a negative stress.

Lithostatic pressure may produce strain in rocks due to simple compaction, but it probably cannot account for most folding and faulting. For this, differential stress is required. Differential stress may be developed by unequal normal stresses or by shear stress—see Figs. 9-1, 9-2.

Strain in rocks is accomplished by dilation, which is a change only in size, or by distortion, which is a pure shape change, or commonly by both. Strain is most easily measured as a percentage length change.

```           change in length
e  =  ---------------- x 100,  where e = percent strain.
original length```
The amount of strain is, of course, related to the magnitude of stress, and the strain response of various rocks to stress may be tested in laboratory apparatus. Prepared rock cores are squeezed in a hydraulic press subjecting the rock to high õ1 compression. Measuring devices attached to the press and rock sample monitor pressure and detect minute strains in the rock. Pressures as high as 100 kb could be produced by uniaxial or triaxial hydralic presses. The highest experimental compression yet achieved is produced by a diamond-anvil technique that has reached pressures of 1700 kb (Jayaraman 1984).

The standard means of portraying the behavior of a stressed rock is with a stress-strain diagram, where strain is plotted on the horizontal axis and stress is plotted on the vertical axis—see Fig. 9-3. Strain is the percentage of shortening parallel to õ1 and stress is the difference õ1 - õ3. As differential stress increases, most rocks initially behave in an elastic manner: strain is directly proportional to stress, and strain disappears when stress is removed. This results in a straight line on the graph.

As differential stress is increased further, the rock reaches the so-called yield strength or elastic limit, beyond which the rock behaves in a plastic manner and the strain does not disappear when stress is removed. Strain increases rapidly with only slightly higher stress until the ultimate strength of the rock is achieved. The rupture strength represents the point at which the rock fails by fracturing. Under conditions of low confining pressure, most rocks fail before reaching the elastic limit, but the same rocks have a higher rupture strength and behave plastically under higher confining pressure.

### Rock failure

The failure of a rock by rupturing results in sets of fractures whose orientations are related to the õ1, õ2, and õ3 stress axes—see Fig. 9-4.

• Shear fractures -- plane of failure oriented about 30° from õ1, 60° from õ3, and parallel to õ2. Most common type; two crossing sets may form.

• Extension fractures -- plane of failure parallel to õ1 and õ2, and perpendicular to õ3. Not common geologically; develop only where there is little or no confining pressure.

• Release fractures -- plane of failure perpendicular to õ1, and parallel to õ2 and õ3. Develop when stress load is released, as when deeply buried rocks are unloaded by erosion of overburden.

Shear fractures are the most common type of geologic fracture, and they develop because of the stress difference between õ1 and õ3. As õ2 is an intermediate stress, it may be ignored and shear fracturing analyzed in two dimensions—see Fig. 9-5A. Consider a plane within the rock which forms an angle (Ð) with the õ1 axis. The õ1 stress operating on the plane is composed of two components: (1) n1, a normal stress at right-angle to the plane, and (2) T1, a shear stress parallel to the plane. Likewise, the normal and shear stress components of õ3 may be determined and added to n1 and T1 to give the total normal stress (nt) and shear stress (Tt) operating on the plane. It is possible to calculate the total normal and shear stresses for a plane of any orientation using the following formulas.

```           õ1 + õ3     õ1 - õ3
n  =  -------  -  ------- (cos 2Ð)
2           2

õ1 - õ3
T  =  ------- (sin 2Ð)
2```
Where õ1 > õ3, shear stress is developed on all planes except those oriented Ð = 0° or 90°, and maximum shear stress is achieved on the plane at Ð = 45°. As rock shear strength is generally much less than compressive strength, shear fractures tend to develop obliquely to the õ1 axis during differential compression.

Rocks subjected to differential compression develop maximum shear stress along planes oriented Ð = 45°. However, Figure 9-4 shows, actual shear fractures typically form at Ð = 30° to 35°. Shear fractures represent the initiation of faulting, and there are two attributes of rocks which resist faulting--cohesive strength and internal friction. Both of these phenomena are demonstrated by dragging a brick over a horizontal surface—see Fig. 9-5B.

In order to drag the brick, a horizontal shear stress must be applied. The total stress operating on the brick is the shear stress plus the normal stress of gravity, and the total stress vector forms an angle (ø) with the vertical. Ø represents the angle of internal friction; its magnitude is a measure of frictional resistance to sliding of the brick. Now, suppose the brick is bonded to the underlying surface by mortar. An additional shear stress must be applied to break the bond before any movement could take place. This additional shear stress, labelled To, is the cohesive strength of the mortar.

### Mohr diagram

The Mohr stress circle provides a convenient graphical means of solving the normal and shear stress equations presented in the previous section. Normal stresses are plotted on the horizontal axis and shear stress read on the vertical axis—see Fig. 9-6. A semicircle with its center on the normal stress axis at (õ1 + õ3)/2 connects õ1 and õ3, and the radius of the semicircle equals (õ1 - õ3)/2. The angle 2Ð is measured from õ3 in clockwise direction.

The point on the semicircle for a given 2Ð angle indicates a normal stress value and a shear stress value for a plane oriented at angle Ð within a rock. The maximum shear stress is achieved at the top of the circle, where 2Ð = 90°, or Ð = 45°. Thus, the maximum shear stress is equal to the radius of the Mohr stress circle. Note that the Mohr circle has no physical reality; it is simply a graphical solution to the stress equations.

A series of stress circles representing different test conditions for a particular rock may be plotted together to produce a Mohr failure envelope—see Fig.9-7. Each Mohr circle represents the õ1 and Ð conditions of shear fracturing for a given confining pressure (õ3). As confining pressure increases, the õ1 stress and Ð angle at which failure occurs also increases. The points of rock failure on each circle define a line that is more-or-less straight at lower confining pressures and flattens out toward higher confining pressures. This line is the failure envelope; its intercept with the shear-stress axis gives cohesive strength (To), and its slope is the angle of internal friction (ø). Any point on or above the failure envelope represents stress conditions that would cause the rock to fracture.

Construction of a Mohr failure envelope represents a powerful means of evaluating rock strength, and a great many failure envelopes have been determined for all kinds of rocks under various temperature and pore-fluid conditions. The angle of friction and cohesive strength vary widely—see Table 9-1. Based on the geometry of the failure envelope, it may be seen that ø is related to 2Ð, such that:

ø = 90 - 2Ð, or Ð = (90 - ø)/2.

As many rocks possess an internal angle of friction of approximately 30°, the Ð angle of shear fractures is typically also about 30°.

Table 9-1. Average values for angle of friction (ø) and cohesive strength (To in bars) in various rock types.
Rock Type Ø To
Igneous: Plutonic 45.6 561
Igneous: Volcanic 24.7 322
Metamorphic: Foliated 27.3 457
Metamorphic: Non-foliated 36.6 229
Sedimentary: Clastic 29.2 317
Sedimentary: Chemical 35.9 263
All Types (average) 32.0 345

Based on Kulhawy (1975, table XI).

### Problem

Andrews (1980) has described glacial thrusting of Paleozoic limestone and shale along Densmore Creek in western New York near Rochester—see Fig. 9-8. A bedrock mass consisting of limestone and weathered shale has been thrust along a zone of brecciated shale containing disoriented angular blocks of limestone—see Fig. 9-9. Below this thrust, till rests on intact Irondequoit Limestone at the western end of the section, and overturned folds are developed in the limestone at the eastern end of the section.

 Genesse River gorge in western New York, near Rochester. A thick sequence of upper Devonian clastic strata is exposed in the walls of the canyon. Photo date 4/74, © J.S. Aber.

The orientations of these folds and nearby striations indicate that ice flow locally turned toward the southwest following a bedrock valley tributary to Irondequoit Bay. The ice margin at the time of thrusting was probably only a few km to the south, so that ice thickness at the site of thrusting was likely no more than 300 m.

A temperate glacier, which is separated from its bed by a film of meltwater, may develop basal shear stress up to 1 bar, and basal shear stress may reach 10 bars for glaciers frozen to the substratum (Weertman 1961). The lithostatic pressure developed at the base of a glacier is given by:

lithostatic P (in kg/cm²) = ice thickness (in m) x 0.09

1. Intact sample-cores of the Irondequoit Limestone have an average unconfined compressive strength of approximately 1200 bars. Construct a Mohr stress circle for the Iron dequoit Limestone.

2. Assume an average ø value of 36° for limestone (chemical sedimentary rock), and plot a straight-line failure envelope on the Mohr diagram. Determine the To and Ð values.

3. Use the stress equations to calculate T and n stress values for unconfined failure of the Irondequoit Limestone. Compare calculated values with graphical results.

4. Assume the glacier was 300 m thick over Densmore Creek when bedrock thrusting took place. Calculate the confining (lithostatic) pressure developed at the base of the ice.

5. Plot the basal stress conditions for both frozen and thawed glacier-bed situations on the Mohr diagram. Are the stresses developed for either type of glacier bed sufficient to fracture intact Irondequoit Limestone?

6. Based on your answer to question 5 and the geological setting (fig. 9-9), explain how ice shoving was able to thrust bedrock at Densmore Creek.

### References

• Andrews, D.E. 1980. Glacially thrust bed-rock--An indication of late Wisconsin climate in western New York State. Geology 8:97-101.

• Davis, G.H. 1984. Structural Geology of Rocks and Regions. J. Wiley & Sons, New York.

• Hobbs, B.E., Means, W.D. and Williams, P.F. 1976. An Outline of Structural Geology. J. Wiley & Sons, New York, 571 p.

• Jayaraman, A. 1984. The diamond-anvil high-pressure cell. Scientific American 250/4:54-62.

• Kulhawy, F.H. 1975. Stress deformation properties of rock and rock discontinuities. Engineering Geology 9:327-350.

• Simpson, C. and Schmid, S.M. 1983. An evaluation of criteria to deduce the sense of movement in sheared rocks. Geological Society of America, Bulletin 94:1281-1288.

• Weertman, J. 1961. Mechanism for the formation of inner moraines found near the edge of cold ice caps and ice sheets. Journal of Glaciology 3:965-978.

Notice: This course is presented for the use and benefit of students enrolled at Emporia State University. Others are welcome to view the course webpages. Any other use of text, imagery or curriculum materials is prohibited without permission of the instructor. © J.S. Aber (2012).