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geological engineering

Design Analysis in Rock Mechanics 1st edition William G. Pariseau - Solutions

Explain the objective of rock mass classification schemes, RMR and Q, why RQD is important to such schemes, what the main components are and what the main differences are. Also explain the differences in support strategy concerning fixed steel sets and yieldable arches.
Derive expressions shear force V, bending moment M and sag w for a "weightless" beam with built-in ends of thickness h, breadth b and span L under a uniform load p. Plot the shear and moment diagrams. Use the sign convention that tension is positive, the x-axis is to the right, the z-axis down, the
Show that the maximum tension σt in a roof beam with built-in ends occurs at the top of the beam at the beam ends and is given by the expression (1/2) pL2/h2.
Show that the maximum sag w max in a roof beam with built-in ends occurs at the center and is given by the expression (1/32) pL4/ (Eh3).
Given that the roof beam factor of safety Ft is the ratio of tensile strength To to peak tensile stress, find an expression for the maximum possible roof span as a function of the factor of safety in the built-in ends case.
Consider a rectangular entry 12 ft high and 22 ft wide that is driven at a depth of 950 ft. The immediate roof layer which is 2.25 ft thick separates from the layers above. Layer properties are E = 3.7 × 106 psi, ν = 0.28, γ = 134pcf, To = 690 psi. Find the maximum tensile stress σt , maximum
A high horizontal stress exists in the vicinity of the excavation given in Problem 5a. The stress is just sufficient to reduce the tensile bending stress to zero. Find the magnitude of this horizontal stress, assuming the beam ends are simply supported.
Consider a rectangular entry 3.66 m high and 6.71 m wide that is driven at a depth of 290 m. The immediate roof layer which is 0.69 m thick separates from the layers above. Layer properties are E = 25.52GPa, ν = 0.28, γ = 21.2kN/m3, To = 4.7 MPa. Find the maximum tensile stress σt , maximum sag
A high horizontal stress exists in the vicinity of the excavation given in Problem 7. The stress is just sufficient to reduce the tensile bending stress to zero. Find the magnitude of this horizontal stress, assuming the beam ends are simply supported.
Consider the geologic column and properties given in Table 5.1 for entries developed on strike at a depth of 1,450 ft (442 m) in strata dipping 8?. Determine the potential for bed separation in the roof rock, that is, find any separation horizons. Mining height is 12 ft (3.7 m).
With reference to Problem 9, determine the maximum possible roof span, given that all roof rock must have a safety factor greater thanone.
A solution mine for nacholite that is processed into soda ash is developed by wells 549 m below the surface in the Piceance basin of Colorado. The mine will produce about 1 million short tons per year initially. Assume the “pay zone” is 27 m thick, dips 5◦ and that dissolvable nacholite is
A method of approximating the support action of rock bolts is to equate the bolt tension to a uniformly distributed load acting upward. If the equivalent bolting pressure is pb, what is the magnitude of the bolting pressure that will reduce the sag in Problem b or 5.7b to zero?
Suppose rock bolt load is 16,400 pounds and bolts are spaced on 5 ft centers in a square pattern. Find the bolting “pressure” pb. What layer thickness is this bolting pressure equivalent to assuming a single layer roof beam sagging under self-weight alone with simply supported ends. Note:
Suppose rock bolt load is 73.5kN and bolts are spaced on 1.5 m centers in a square pattern. Find the bolting “pressure” Pb. What layer thickness is this bolting pressure equivalent to assuming a single layer roof beam sagging under self-weight along with simply supported ends? Note: Specific
With reference to the sketch, a bolting plan consisting of 5/8 in. (1.59 cm) diameter mechanically point anchored bolts spaced on 4 ft centers (1.22 m), square pattern, is proposed. Evaluate this plan from the dead weight view. Show why the plan is adequate or not, assume high strength. Would 3/4
Suppose a suspension action is induced by 3/4 in. (1.91 cm) bolts on 5 ft (1.52 m) centers (high strength) installed in the roof shown in the sketch. If the sag is then 40% of the unsupported sag of the suspended strata, what is the reduction in roof tension? Assume that the bolts are tensioned to
With reference to Problem 16, what is the ratio of the supported to unsupported roof rock safety factors?
If the four shale layers in the sketch are clamped such that they deform as a single beam what is the maximum shear stress, and where does it occur in the clamped roof? In this regard, if the coefficient of friction between shale strata is 0.35, what bolting pressure is required to mobilize
Consider the roof bolt truss shown in the sketch. If the spacing of the trusses along the entry is S and the tension in the “bolts” is T, what is the equivalent bolting pressure capacity of the truss system?
With reference to the sketch and Problem 5.19, prescribe S and T such that the apparent unit weight of the laminated shale?s is 65% of the unsupported apparent unit weight.
With reference to the sketch, show in detail the determination of the optimum bolting angle using the givennotation.
With reference to the sketch, suppose the joint cohesion is zero while the friction angle is 35?. If the force per bolt is 12,000 pounds (53.8kN) and the area of influence of a bolt is 4 ft2, find: (a) The unbolted joint safety factor, and (b) The bolted joint safety factor at the optimum angle.
With reference to Problem 22, suppose a passive bolt, un-tensioned and 7/8 in. (2.22 cm) in diameter is grouted in the hole at the optimum angle. Estimate the shear stress in the bolt.
Coal is mined in two seams that are shown in the sketch. Mining is full seam height in both seams. Entries are 24 ft wide (7.3 m).With reference to the stratigraphic column and rock properties in the sketch, determine for the lower seam: (a) The safety factor with respect to tension of the
Both seams of Problem 5.24 are mined by the long wall method. There are five main entries and sub-main entries used for development in both seams. A three entry system is used for panel development. The sub-main entries are at right angles to the mains, while the panel entries are at right angles
A thick seam of coal is mined from the floor to a height of 13 ft at a depth of 1,450 ft. With reference to the stratigraphic column in the sketch and the rock properties given in Table 5.2, assume L=21 ft, then find: (a) The safety factor with respect to tension of the immediate roof layer (b) The
With reference to the strata data given in the sketch and Table 5.3, seam depth is 1670 ft (509 m), strata dip is negligible and entries are planned 24 ft (7.32 m) wide. Mining height from the sandstone floor is 12 ft (3.66 m). 1 Determine the immediate roof safety factor. 2 Determine bolting
With reference to the strata data given in the sketch and Table 5.4, seam depth is 1,324 ft (404 m), strata dip is negligible and entries are planned 21 ft (6.4 m) wide. Mining height from the sandstone floor is 15 ft (4.6 m). 1. Determine the immediate roof safety factor. 2. Determine bolting
Given conditions of Problem 29, if it were feasible to measure roof sag as entries were mined 15 ft wide,(a) How much sag would be observed at mid-span?(b) What sag measurement would indicate impending roof failure at the same span?
With reference to Problems 29 and 30, layout a reasonable bolting plan using a dead weight approach and point anchored mechanical bolts assuming entries and crosscuts are mined 20 ft (6.1 m) wide. Specify bolt length, diameter, spacing, steel grade, and pretension.
A thin "low" metallurgical coal seam 3 ft thick is mined at a depth of 980 ft. The main entries are mined 4 ft into the R1 roof stratum to give sufficient clearance for track haulage. With reference to the stratigraphic column in the sketch and the rock properties given in Table 5.7, find: (a) The
The immediate roof behind a long wall face often considered to be a cantilever beam as shown in section in the sketch. Beginning with a safety factor design criterion, derive an expression for the distance X the face can advance before R1 failure. Assume bed separation between R1 and R2 occurs,
Given the geologic column, strata depths and rock properties in Table 5.8, find: (a) The maximum possible roof span when mining the lower coal seam full height, (b) The safe roof span, if a safety factor of 4 with respect to tension is required, (c) The factor of safety of the massive sandstone
Prescribe a bolting plan for entries driven 28 ft (8.5 m) wide in the lower coal seam in the sketch using the dead weight load approach. Include nominal bolt diameter, length, grade of steel, installation tension, and spacing. Support your recommendation withcalculations.
A regular room and pillar method is used in a hard rock lead–zinc mine. The ore horizon is at a depth of 1,180 ft where the overburden averages 159pcf. Laboratory tests on core samples (L/D = 2) show that the ore has an unconfined compressive strength of 13,700 psi and a tensile strength of 1,250
With reference to Problem 1, if pillars are square and rooms (entries and crosscuts) are 45 ft wide, what size of pillars is indicated?
With reference to Problems 1 and 2, suppose the pillars are 45 × 45 ft and that a size effect exists such that Cp = C1 (0.78 + 0.22 Wp/Hp). If the mining height changes from 30 to 120 ft, as a sill between two ore horizons is removed, what is the resulting pillar safety factor, all other
Suppose that entries and crosscuts are 45 ft wide and that pillars are square and 45 ft wide, as in Problem 6.3. Laboratory properties are the same as in Problem 6.1; depth and specific weight of overburden are also the same as in Problem 6.1. In addition, a joint set is present in the mine. Joint
A regular room and pillar method is used in a hard rock lead–zinc mine. The ore horizon is at a depth of 360 m where the overburden averages 25.2kN/m3. Laboratory tests on core samples (L/D = 2) show that the ore has an unconfined compressive strength of 94.5 MPa and a tensile strength of 8.63
With reference to Problem 5, if pillars are square and rooms (entries and crosscuts) are 13.7 m wide, what size of pillars is indicated?
With reference to Problems 5 and 6, suppose the pillars are 13.7 × 13.7 m and that a size effect exists such that Cp = C1 (0.78 + 0.22Wp/Hp). If the mining height changes from 9.14 to 36.6 m, as a sill between two ore horizons is removed, what is the resulting pillar safety factor, all other
Suppose that entries and crosscuts are 13.7 m wide and that pillars are square and 13.7 m wide, as in Problem 6.7. Laboratory properties are the same as in Problem 6.5; depth and specific weight of overburden are also the same as in Problem 6.5. In addition, a joint set is present in the mine.
Suppose that the average vertical pillar stress is 26.7 MPa in a room and pillar mine and that a joint set is present with an average bearing of N45W. If the joints obey a Mohr–Coulomb slip criterion and have a friction angle of 28◦ and a cohesion of 0.90 MPa, what range of joint dips is unsafe
Coal mine entries are driven 20 ft (6.1 m) wide on strike at a depth of 1200 ft (366 m) in a coal seem that dips 15◦. Crosscuts are driven up and down dip and are also 20 ft (6.1 m) wide. The production pillars that are formed are twice as long in the strike direction as they are in the dip
A room and pillar trona mine at a depth of 1,560 ft (475 m) develops pillars 30 ft (9.14 m) wide and 60 ft (18.3 m) long in a flat seam 12 ft (3.66 m) thick. Entries and crosscuts are 24 ft (7.32 m) wide. Laboratory tests of trona core 2-1/8 in. (5.4 cm) in diameter and 4-1/4 in. (10.8 cm) long
A narrow, 20 ft (6.10 m) thick vein dips 60◦. Stopping occurs at a depth of 5,100 ft (1,555 m) in rock that averages 165pcf (26.1kN/m3). The vertical stress before mining is equal to the unit weight of overburden times depth; the horizontal premining stress is twice the vertical stress.
If mining in Problem 12 is by conventional overhand stopping, and round length (measured on the dip) is 15 ft (4.57 m), is failure likely to be stable or not as mining proceeds past the maximum extraction ratio previously calculated? Explain your answer.
Consider a large array of similar pillars on a regular grid that have a safety factor with respect to compressive failure of Fc. Suppose a single pillar fails and the load is then shared equally by the nearest neighboring pillars.(a) Find the safety factor Fc needed to avoid failure of the nearest
A regular room and pillar method is used in a hard rock lead–zinc mine. The ore horizon is at a depth of 980 ft where the overburden averages 156pcf. Laboratory tests on core samples (L/D = 2) show that the ore has an unconfined compressive strength of 12,400 psi and a tensile strength of 1,050
With reference to Problem 15, if pillars are square and rooms (entries and crosscuts) are 30 ft wide, what size of pillars are indicated when the pillar safety factor is 2?
With reference to Problems 6.15 and 6.16, suppose the pillars are 30×30 ft and that a size effect exists such that Cp = C1 (0.78+0.22 Wp/Hp). If the mining height changes from 25 to 90 ft as a sill between two ore horizons is removed, what is the resulting pillar safety factor, all other
Suppose that entries and crosscuts are 30 ft wide and that pillars are square and 30 ft wide, as in Problem 6.17. Laboratory properties are the same as in Problem 6.15; depth and specific weight of overburden are also the same as in Problem 6.15. In addition, a joint set is present in the mine.
A regular room and pillar method is used in a hard rock lead–zinc mine. The ore horizon is at a depth of 300 m where the overburden averages 25kN/ms. Laboratory tests on core samples (L/D = 2) show that the ore has an unconfined compressive strength of 86 MPa and a tensile strength of 7 MPa. Find
With reference to Problem 19, if pillars are square and rooms (entries and crosscuts) are 9 m wide, what size of pillars is indicated when the pillar safety factor is 2?
With reference to Problems 19 and 20, suppose the pillars are 9 × 9 m and that a size effect exists such that Cp = C1 (0.78 + 0.22Wp/Hp). If the mining height changes from 8 to 27 m as a sill between two ore horizons is removed, what is the resulting pillar safety factor, all other parameters
Suppose that entries and crosscuts are 9 m wide and that pillars are square and 9 m wide, as in Problem 6.21. Laboratory properties are the same as in Problem 6.19; depth and specific weight of overburden are also the same as in Problem 6.19. In addition, a joint set is present in the mine. Joint
Suppose that the pillar stress is 4,247 psi in a room and pillar mine and that a joint set is present with an average bearing of N45W. If the joints obey a Mohr–Coulomb slip criterion and have a friction angle of 35◦ and a cohesion of 600 psi, what range of joint dips is unsafe with respect to
Suppose that the pillar stress is 29 MPa in a room and pillar mine and that a joint set is present with an average bearing of N45W. If the joints obey a Mohr–Coulomb slip criterion and have a friction angle of 35◦ and a cohesion of 4 MPa, what range of joint dips is unsafe with respect to joint
Coal is mined in two seams that are shown in the sketch. Mining is full seam height in both seams. Entries are 24 ft (7.3 m) wide. Crosscuts associated with driving main entries are 18 ft (5.5 m) wide and driven on 120 ft (36.6 m) centers. Associated pillar length is three times pillar width. With
A pillar in a hard rock, strata-bound, room, and pillar mine is transected by a small fault that dips 75? due east and strikes due north. The ore horizon dips due west 15? and also strikes due north. Depth to the pillar of interest is 1,230 ft (375 m). Overburden specific weight averages 158pcf
A thick seam of coal is mined from the floor to a height of 13 ft at a depth of 1450 ft. Entry width is 21 ft and overburden averages 144pcf. Crosscuts are 18 ft wide and driven on 100 ft centers. Pillar length is three times pillar width. With reference to the stratigraphic column in the sketch
A pillar in a hard rock, strata-bound, room, and pillar mine is transected by a small fault that dips 75? due east and strikes due north. The ore horizon dips due west 15? and also strikes due north. Depth to the pillar of interest is 1,230 ft (375 m). Overburden specific weight averages 158pcf
With reference to the strata data given in the sketch and Table 6.4, seam depth is 1,670 ft, strata dip is negligible, and entries are planned 24 ft wide. Mining height from the sandstone floor is 12 ft. If crosscuts are as wide as entries and pillar length is three times pillar width:1. Determine
Several proposals have been presented to increase the extraction ratio in a room and pillar system to 50% from 33%. The pillars are long compared with width, so the problem is essentially two-dimensional. Proposal 1; Increase the ratio of opening width to pillar width from 0.5 to 1.0 by
Several proposals have been presented to increase the extraction ratio in a room and pillar system to 50% from 33%. The pillars are long compared with width, so the problem is essentially two-dimensional.Proposal 1; Increase the ratio of opening width to pillar width from 0.5 to 1.0 by increasing
A multilevel room and pillar metal mine is under consideration in flat strata striking N60E. Entries are planned on strike, crosscuts up, and down dip. Table 6.6 shows depths and material properties associated with the geologic column. Rock properties data were determined from laboratory testing on
With reference to Problem 6.32 data, roof rock above the upper A Level is Bonneterre dolomite that is overlain by Davis shale.(a) Determine the maximum opening width that is physically possible in the upper A Level ore horizon.(b) With reference to Problem 6.32, assume square pillars, then
With reference to Problem 32, concerning opening width in the A Level, determine a bolting plan that would allow for 90 ft wide rooms.(a) Specify, bolt length, safety factor, diameter, steel grade, and spacing.(b) Also specify the associated pillar size and pillar safety factor with respect to ore
An alternative mining plan is to develop two levels, below the upper A Level, and mine only the high grade. The B Level would be in the 43 ft of high grade, while the C level would be in the lower 20 ft of high grade in the 102 ft column shown in the detailed geologic column. Determine the maximum
With reference to Problem 6.32, if pillars are sized according to the maximum extraction ratio and pillar compressive strength, determine if the pillars are safe with respect to failure of Joint Set 2.
A multilevel room and pillar metal mine is under consideration in flat strata striking N60E. Entries are planned on strike, crosscuts up, and down dip. Table 6.8 shows depths and material properties associated with the geologic column. Rock properties data were determined from laboratory testing on
With reference to Problem 37 data, roof rock above the upper A Level is Bonneterre dolomite that is overlain by Davis shale.(a) Determine the maximum opening width that is physically possible in the upper A Level ore horizon.(b) With reference to Problem 6.37, assume square pillars, then determine
With reference to Problem 38, concerning opening width in the A Level, determine a bolting plan that would allow for 27.4 m wide rooms.(a) Specify, bolt length, safety factor, diameter, steel grade, and spacing.(b) Also specify the associated pillar size and pillar safety factor with respect to ore
An alternative mining plan is to develop two levels, below the upper A Level and mine only the high grade. The B Level would be in the 13.1 m of high grade, while the C level would be in the lower 6.10 m of high grade in the 31.1 m column shown in the detailed geologic column. Determine the maximum
A room and pillar coal mine is contemplated at a depth of 533 m in strata striking due north and dipping 18? east. Entries are driven on strike, crosscuts up, and down dip. Mining height is 4.6 m measured from the floor; 0.5 m of low grade coal is left in the roof. Three joint sets are present. Set
With reference to the data given in Problem 6.42, if a joint safety factor of 1.75 with respect to shear is required, what extraction ratio is possible considering only joint Set 3 as important? What is the maximum possible extraction ratio allowed by joint Set 3 comment?
With reference to the data in Table 6.10, occasionally, caving to the coal ?rider? seam just below the limestone stratum happens at intersections of the 20 ft wide entries and crosscuts. Specify a cable bolt plan that will prevent these caves assuming the steel?grout interface bond strength is 750
After grouting, a cemented fill, that has completely filled the old workings, has a Young’s modulus of 100ksi (690 MPa), and an unconfined compressive strength of 3,400 psi (23.4 MPa), as determined from unconfined compressive strength tests. The original extraction ratio in this area is
A thin ?low? metallurgical coal seam 3 ft (1 m) thick is mined at a depth of 980 ft (300 m). The main entries are mined 4 ft into the R1 roof stratum to give sufficient clearance for track haulage. With reference to the strati graphic column in the sketch and the rock properties given in Table
Main entries are driven in sets of seven at a depth of 1,530 ft in flat-lying strata. Entries are 20 ft wide (Wo); pillars are 80 ft wide (Wp). In vertical section showing entries and pillar widths, the two-dimensional extraction ratio R' = Wo/(Wo + Wp). Three proposals are presented for increasing
Main entries are driven in sets of seven at a depth of 1,466 mt in flat-lying strata. Entries are 6 m wide (Wo); pillars are 24 m wide (Wp). In vertical section showing entries and pillar widths, the two-dimensional extraction ratio R' = Wo/ (Wo + Wp). Three proposals are presented for increasing
Development entries 16 ft (4.9 m) wide are driven in a steeply dipping anthracite coal seam at a depth of 760 ft (232 m). Crosscuts are not a factor with distance between raises on the dip at 300 ft (91 m). (The extraction ratio in cross section is thus R'.) Strata dip is 55◦; seam thickness is
Consider an underground limestone mine in flat strata where pillars are 95 ft high and depth is 845 ft. Rock properties are: E = 11.3 million psi, v = 0.20, Co = 21, 500 psi, To = 1, 680 psi, unit weight = 156pcf. A joint set of variable dip pervades the mine. Mohr–Coulomb properties are: c = 630
Consider an underground limestone mine in flat strata where pillars are 29 m high and depth is 258 m. Rock properties are: E = 77.9GPa, v = 0.20, Co = 148 MPa, To = 11.6 MPa, unit weight = 24.7kN/m3. A joint set of variable dip pervades the mine. Mohr–Coulomb properties are: c = 4.34 MPa, φ =
With reference to the sketch, Table 6.13, and the lower coal seam, what is the maximum possible extractionratio?
With reference to the sketch, Table 6.13 and the lower coal seam, if a factor of safety of 1.5 with respect to compression is required when entries and crosscuts are actually 18 ft (5.5 m) wide and pillars are twice as long as they are wide, find (with neglect of any size effect on strength): 1
A tabular ore body is 32 ft (9.75 m) thick, strikes due north, and dips 22◦ east. The premining stress field is due to gravity alone. Stopes are excavated at a depth of 2,590 ft (179 m) in a regular room and pillar configuration. A lane and pillar method is used, so the system appears
With reference to the data given in Problem 54, suppose that a joint system is present. If the joints also strike due north and dip to the 45◦ east, find the joint cohesion needed for a safety factor or 1.5 with respect to joint shear. Note: the joints are clay-filled and have a friction angle of
An oblate spheroid is solution mined in a salt bed with a = 75 ft and b = 150 ft. Center of the cavity is 650 ft deep. Estimate peak stresses and show locations on a cavity sketch.
An oblate spheroid is solution mined in a salt bed with a = 23 m and b = 46 m. Center of the cavity is 200 m deep. Estimate peak stresses and show locations on a cavity sketch.
An underground storage cavern is solution mined in a thick salt bed whereCo = 13,200 psi To = 1,230 psiγ = 144pcf E = 5.0 × 106 psiG = 2.0 × 106 psiand depth to the cavern center is 1,450 ft. The cavern has a spherical shape and is 150 ft in diameter. The premining stress is hydrostatic.
An underground storage cavern is solution mined in a thick salt bed whereCo = 91.0 MPa To = 8.5MPaγ = 22.8kN/m3 E = 34.5GPaG = 13.8GPaand depth to the cavern center is 442 m. The cavern has a spherical shape and is 49 m in diameter. The premining stress is hydrostatic. Estimate the wall safety
An underground storage cavity is excavated in a massive salt formation by solution mining. Depth to the center of the cavity is 1,340 ft (408 m). Borehole surveying shows the cavity has the shape of an oblate spheroid 100 ft (30.5 m) in diameter and 50 ft (15.25 m) high. Estimate the salt strength
Mining over the years produces a tabular excavation that extends about 2,000 ft (610 m) down dip from the surface. Dip of the ore zone is 65◦, width is 40 ft (12.2 m). Strike length is over 6,000 ft (1,839 m). Estimate the peak stresses and indicate their location on a sketch of the mine.
Stopes mined along plunging folds of Precambrian rock range between depths of 869 m (2,850 ft) and 1,174 m (3,850 ft) below surface. The stopes are up to 30.5 m (100 ft) wide. Fold plunge is 12◦. Fold dip varies, but stope walls are vertical. A vertical, rectangular shaft 4.6 × 6.4 m (15 × 21
Consider a cut and fill stope in a 15 ft wide vein where raise and level intervals are 175 ft. A hydraulic fill is placed along the entire stope length to a height of 12 ft. If the specific weight of fill is 100pcf and the porosity is 35%, how many tons of solids are in the fill and how many
Consider a cut and fill stope in a 4.6 m wide vein where raise and level intervals are 53.3 m. A hydraulic fill is placed along the entire stope length to a height of 3.66 m. If the specific weight of fill is 15.8kN/m3 and the porosity is 35%, how many tons of solids are in the fill and how many
Void ratio e is defined as the ratio of void volume to volume of solids in porous material. Show that void ratio may be computed from porosity by the formula n/(1−n) where n is porosity.
Consider a narrow vein overhand stope using cut and fill, as shown in the sketch. Cemented, hydraulic sand fill of modulus Ef is placed in the stope as mining proceeds up dip in a rock mass of modulus Er.? (a) Develop a formula for crown pillar safety factor assuming uniform closure across the

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