Assignment 4
(Due: 17 March, 2020 by 17:00)
Q1. A rock slope for an open pit mine is 60 m high and has a face angle of 60°. The main discontinuity plane dips at an angle of 40° into the excavation. The Mohr-Coulomb strength parameters along the joint surfaces are: c = 345 kPa and f = 30°, and the unit weight of the rock mass is 24 kN/m3. Assuming that a plane slope failure is the most likely type of instability, analyze the following stability conditions:
(a). Calculate the factor of safety of the slope for the conditions given in Figure 1a (no surcharge, dry slope and no tension crack);
(b). Determine the maximum height, H to which the slope in part a) can be excavated
without causing failure;
(c). Calculate the factor of safety if the dry slope is reinforced with rock bolts (Figure 1b) with ultimate strength of 2.2 MN. The bolts are installed at an angle of 5° below horizontal, and the bolt spacing is 10 m in vertical and 9 m in horizontal direction. Design tension in the bolts is 60% of the ultimate strength.
(d). A uniformly distributed surcharge, q is to be applied to the dry slope crest over an area (BL), where L = 7 m. Determine the magnitude of the surcharge in kN/m2 that can be applied for a FS = 1.3 against planar failure (Figure 1c);
(e). A uniformly distributed surcharge of 1700 kN/m2 is to be applied to the crest of the dry slope in Figure 1d. The slope will be reinforced with rock bolts with ultimate strength of 2.2 MN, installed at an angle of 5° below horizontal. Design tension in the bolts is 60% of the ultimate strength. Suggest a bolt layout, that is, the number of bolts per vertical row, and the horizontal and vertical spacing between bolts to achieve a FS = 1.3 against planar failure.
(f). The slope in Figure 1e is excavated with a 10 m-wide bench at ½ of the slope height. Consider a tension crack at the crest of the slope and calculate its position behind the crest, B and its depth, z. Determine the factor of safety if the slope was dry and the sliding plane terminated in the tension crack.
(g). During a rainfall event the ground water table rises to a depth of 3 m below the crest of the slope in Figure 1d. Calculate the new factor of safety for the saturated slope.
a) b)
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c) d)
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e)
Figure 1. Slope and failure plane geometry (Plane failure)
Q2. A 60 m–high slope is to be excavated at a face angle of 45° in a blocky sandstone rock mass with very closely spaced and persistent discontinuities. The unit weight of the sandstone is 25 kN/m3 and the Mohr-Coulomb strength parameters along the potential failure plane are: c = 145 kPa and f = 43°. Find the factor of safety of the slope, assuming:
(a). there is a surface water source 240 m behind the toe of the slope; (b). the slope is fully saturated.
(c). For the fully saturated slope in part b), find the locations of the critical slip surface through the toe and of the tension crack.
Q3. A rectangular block of rock is tied to a 40°- slope using cable ties. The block rock material has a unit weight of 26 kN/m3 and dimensions of 25 m x 14 m x 6 m (Figure 2). The angle of friction between the rock block and the slope is 34°. Each cable tie has a diameter of 32 mm and is tightened to a tension of 690 MPa.
(a). How many cable ties are needed to provide a factor of safety against sliding equal to 1.3? Neglect cohesion on the sliding plane.
(b). With the cable ties installed and neglecting cohesion on the sliding plane, find the smallest external driving force that will cause the block to slip.
(c). If an external force of 5 MN is applied parallel to and down the unreinforced slope, what is the magnitude of cohesion that has to develop on the failure plane to just prevent sliding.
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Figure 2. Slope and failure plane geometry (sliding failure)