Questions and Answers of Geological Engineering

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Given a cylindrical sandstone laboratory test specimen of diameter D and height L subjected to a vertical load F (force), find (a) Te forces N and

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Given a rectangular prism of edge length D and height L subjected to a vertical load F (force) applied to the prism ends and a horizontal pressure p

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Is the following equation a physical law, kinematic relationship, or material law? Ms = ∫v rdm Here M =mass in volume V, dm = a mass

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Is the following equation a physical law, kinematic relationship, or material law? ˙σ = ∂σ / ∂t

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Darcy’s law for fluid flow in porous media relates the seepage velocity v to the hydraulic gradient h through a constant k (hydraulic

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Consider a “two-dimensional” state of stress in the x–y plane characterized by: σxx = 2,500 σyy = 5,200 τxy = 3,700

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Given the stress state in Problem 1.11, find: the magnitude and direction of the maximum shear stress and illustrate with a sketch.

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Consider a “two-dimensional” state of stress in the x–y plane characterized by: σxx = 17.24 σyy = 35.86 τxy = 25.52 Where units

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Given the stress state in Problem 1.13, find: the magnitude and direction of the maximum shear stress and illustrate with a sketch

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Suppose that σxx = 2,500 σyy = 5,200 τxy = 3,700 and the z-direction shear stresses (τzx, τzy) are zero, while the

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With reference to Problem 1.15, find the magnitude of the maximum shear stress and show by sketch the orientation of the plane it acts on.

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Suppose that σxx = 17.24 σyy = 35.86 τxy = 25.52 in MPa and the z-direction shear stresses (τzx, τzy) are zero, while the

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With reference to Problem 1.17, find the magnitude of the maximum shear stress and show by sketch the orientation of the plane it acts on.

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Consider a “two-dimensional” state of stress in the x–y plane characterized by: σxx = 5,200 σyy = 2,500 τxy = −3,700 where

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Consider a “two-dimensional” state of stress in the x–y plane characterized by: σxx = 35.86 σyy = 17.24 τxy = −25.52

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Consider a three-dimensional state of stress characterized by: σxx = 3,000 σyy = 2,000 τxy = 0. σzz = 4,000 τzx = 0.

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Given the stress state in Problem 1.21, find the state of stress relative to axes abc(σaa, . . . , τca) rotated 30◦ counterclockwise

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Consider a three-dimensional state of stress characterized by: σxx = 20.69 σyy = 13.79 τxy = 0.0 σzz = 27.59 τzx = 0.0

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Given the stress state in Problem 1.23, find the state of stress relative to axes abc(σaa, . . . , τca) rotated 30◦ counterclockwise

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Show in two dimensions that the mean normal stress σm = (σxx +σyy)/2 is invariant under a rotation of the reference axes and is equal

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Show in two dimensions that the maximum shear stress τm = {[(σxx − σyy)/2]2 + (τxy)2}(1/2) is invariant under a rotation

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Consider an NX-size drill core (2-1/8 in., 5.40 cm diameter) with an L/D ratio (length to diameter) of two under an axial load of 35,466 pounds

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Strain measurements are made on a wide, flat bench in a dimension stone quarry using a 0-45-90 rosette. The 0-gauge is oriented N60E. Specific weight

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Given the stresses in psi: σxx = 1,500 σyy = −2,000 σzz = 3,500 τxy = 600 τyz = −300 τzx =

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Given the stresses in MPa: σxx = 10.35 σyy = −13.79 σzz = 24.14 τxy = 4.14 τyz = −2.07 τzx =

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Given the strains: εxx = 2,000 εyy = 3,000 εzz = 4,500 γxy = −200 γyz = 300 γzx = 225 Where x = east, y =

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Consider a cylindrical test specimen under a confining pressure of 3,000 psi and an axial stress of 3,000 psi with compression positive, so that in

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Given the strains: εxx = 2,000 εyy = 3,000 εzz = 4,500 γxy = −200 τyz = 300 γzx = 225 Where x = east, y =

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Consider a cylindrical test specimen under a confining pressure of 20.69 MPa and an axial stress of 20.69 MPa with compression positive, so that in

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Show that under complete lateral restraint and gravity loading only, that the vertical normal stress in a homogeneous, isotropic linearly elastic

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Suppose an NX-size test cylinder with an L/D ratio of two has a Young’s modulus of 10 million psi, a Poisson’s ratio of 0.35 and fails in

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Suppose an NX-size test cylinder with an L/D ratio of two has a Young’s modulus of 68.97 GPa, a Poisson’s ratio of 0.35, and fails in uniaxial

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Consider gravity loading under complete lateral restraint in flat, stratified ground where each stratum is homogeneous, isotropic, and linearly

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Consider gravity loading under complete lateral restraint in flat, stratified ground where each stratum is homogeneous, isotropic, and linearly

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Consider gravity loading only under complete lateral restraint in flat strata with properties given in Table 1.2. Vertical stress at the top of the

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Consider gravity loading only under complete lateral restraint in flat strata with properties given in Table 1.3. Vertical stress at the top of the

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A stress measurement made in a deep borehole at a depth of 500 m (1,640 ft) indicates a horizontal stress that is three times the vertical stress and

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Identify the three major categories of equations that form the “recipe” for calculating rock mass motion. Give an example of each in equation

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If ρ is mass density, explain why or why not the conservation of mass may be expressed as M = ∫v ρdV For a body of mass M occupying

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Name the two types of external forces recognized in mechanics and give an example of each.

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A huge boulder is inadvertently cast high into the air during an open pit mine blast. Using the definition of mass center and Newton’s second law

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Suppose a static factor of safety is defined as the ratio of resisting to driving forces, that is, FS = R/D. Show that a factor of safety less than

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Derive an expression for the factor of safety FS for a planar block slide with a tension crack behind the crest. The slope height is H and has a face

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Modify the expression for the safety factor from Problem 2.1 to include the effect of a water force P acting on the inclined failure surface. Note

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With reference to Problem 2.2, suppose the water pressure p increases linearly with depth (according to p = γwz where z = depth below water table

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Consider a seismic load effect S that acts horizontally through the slide mass center with an acceleration as given as a decimal fraction a0 of the

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Given the planar block slide data shown in the sketch where a uniformly distributed surcharge σ is applied to the slope crest over an area (bl),

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Consider the planar block slide in the sketch. If no surcharge is present, find the maximum depth H of excavation possible before failure impends.

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Suppose the slope in the sketch is cable bolted. No surcharge is present. Bolt spacing is 50 ft (15.2 m) in the vertical direction and 25 ft (7.6 m)

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Consider the planar block slide in sketch without surcharge, seismic load, and bolt reinforcement and suppose the water table rises to the top of the

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Suppose the cohesion of the slide mass shown in the sketch decreases to zero and no surcharge, seismic load, or water is present. Determine whether

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Some data for a possible planar block slide are given in the sketch. If α = 35◦, β = 45◦, and H = 475 ft (145 m), what cohesion c (psf, kPa) is

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Consider the planar block slide in the sketch. (a) Show algebraically that reduction of the slide mass volume from Vo to V1 by excavating a

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With reference to the sketch, if no tension crack is present, α = 29◦, γ = 156pcf (24.7kN/m3), joint persistence = 0.87, no crest relieving

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A generic diagram of a slope in a jointed rock mass that is threatened by a planar block slide is shown, in the sketch. Although not shown, bench

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A generic diagram of a slope in a jointed rock mass that is threatened by a planar block slide is shown in the sketch. A minimum safety factor of

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Given the following planar block slide data: Mohr–Coulomb failure criterion applies1 slope height

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A generic diagram of a slope in a jointed rock mass that is threatened by a planar block slide is shown in the sketch. With neglect of any seismic

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With reference to the sketch of the potential slope failure shown in the sketch, find:

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Given the data in Table 2.7, determine the direction angles of the normal vectors to joint planes A andB.

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Calculate the dip and dip direction of the line of intersection of joint planes A and B, then sketch the result in compass coordinates.

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Given that the vertical distance between points a and d is 120 ft (36.6 m), determine the areas of joint planes A and B, assuming no tension crack is

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Use computer programs to verify the area data in Table 2.7 with a tension crack present and a rock specific weight of 158pcf (25.0kN/m3). What is the

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With reference to the generic wedge in the sketch and the specific data in Table 2.8 if the safety factor with respect to sliding down the line of

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Two joints K1 and K2 are mapped in the vicinity of a proposed surface mine. Joints in set K1 have dip directions of 110◦ and dips of 38◦; joints

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With reference to the generic wedge shown in the sketch and the data in Table 2.9, the water table is below the toe, vertical distance between A and

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With reference to the wedge shown in the sketch and the Table 2.10 data, if the safety factor with respect to sliding down the line of intersection

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Given the wedge data in Table 2.11, if the vertical distance between crest and the point where the line of intersection between the joint planes

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With reference to the generic wedge shown in the sketch and the data in Table 2.12, the water table is below the toe, vertical distance between A and

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Consider a bank 40 ft (12.2 m) high with a slope of 30◦ that may fail by rotation on a slip circle of radius 60 ft (18.3 m) with center at (20, 48

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With reference to the sketch and tables, find the safety factors for the four cases posed using the method of slices. Include all important

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With reference to the sketch illustrating a rotational slide being considered for a method of slices stability analysis, show that the inclusion of

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Given the potential one-foot thick slope failure along the circular arc (R = 300 ft, 91.44 m) shown in the sketch, the data in Table 2.14, a slope

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Given the potential one-foot thick slope failure along the circular arc (R = 300 ft, 91.44 m) shown in the sketch and the data in Table 2.14, find

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Given the potential one-foot thick slope failure along the circular arc shown in the sketch and the data in Table 2.14, find the safety factor for

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Given the circular failure illustrated in the sketch, find algebraically the factor of safety. Assume that the material properties and geometry of

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Consider the possibility of a rock slide starting at point A and ending at B, as shown in the sketch, and suppose the slide is driven by gravity and

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Given the slope profile in vertical section shown in the sketch and an angle of friction φ = 15◦, if the mass center of the slide is initially at

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Toppling of tall blocks is possible when one corner tends to lift off from the base supporting the block, as shown in the sketch. Show that the block

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A rock block with a square base rests on a 28◦ slope. The block base plane has a friction angle of 32◦. There is no adhesion between block and

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A 20 foot diameter circular shaft is planned in a massive rock. Laboratory tests on core from exploration drilling show thatCo = 22,000 psi

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If the opening in Problem 1 is in an in situ stress field such that σh = σv, what are the safety factors?

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If the opening in Problem 1 is in an in situ stress field such that σh = 0.01σv, is there a possibility of failure? Justify your answer.

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A 6 m diameter circular shaft is planned in massive rock. Laboratory tests on core from exploration drilling show thatCo = 152 MPa To = 8.3MPaγ =

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If the opening in Problem 4 is in an in situ stress field such that σh = σv, what are the safety factors?

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If the opening in Problem 4 is in an in situ stress field such that σh = 0.01σv, is there a possibility of failure? Justify your answer.

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Which is preferable from the rock mechanics viewpoint: an elliptical, ovaloidal, or rectangular opening of Wo/Ho of 2 for a stress field M = 1/3? M =

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A rectangular shaft 10 ft wide by 20 ft long is sunk vertically in ground where the state of premining stress is σxx = 1,141, σyy = 2,059, σzz =

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A rectangular shaft 3 m wide by 6 m long is sunk vertically in ground where the state of premining stress is σxx = 7.9, σyy = 14.2, σzz = 11.0,

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A 12 ft by 24 ft rectangular shaft is sunk to a depth of 3,000 ft in ground where the premining stress field is given by formulas Sv = 1.2h, Sh = 120

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A 3.66 m by 7.32 m rectangular shaft is sunk to a depth of 914 m in ground where the premining stress field is given by formulas Sv = 27.2h, Sh = 826

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Thin overburden is scrapped to expose fresh bedrock at the site of a planned shaft. A 0-45-90 strain gauge rosette is bonded to the bedrock after a

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With reference to Problem 12, if the state of stress at the surface continues to depth in addition to gravity load, and the specific weight of rock

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Thin overburden is scrapped to expose fresh bedrock at the site of a planned shaft. A 0-45-90 strain gauge rosette is bonded to the bedrock after a

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With reference to Problem 14, if the state of stress at the surface continues to depth in addition to gravity load, and the specific weight of rock

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With reference to Problem 14, if the state of stress at the surface continues to depth in addition to gravity load, and the specific weight of rock

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If the premining stress state in Problem 16 where hydrostatic (3D), what shape would be more favorable, an ellipse, rectangle, or ovaloid (with

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If the premining stress state in Problem 16 where hydrostatic (3D), what shape would be more favorable, an ellipse, rectangle, or ovaloid (with

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If the premining stress state in Problem 18 where hydrostatic (3D), what shape would be more favorable, an ellipse, rectangle, or ovaloid (with

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A rectangular shaft 18 ft by 24 ft is planned for a depth of 4,800 ft where the premining stresses relative to compass coordinates (x = east, y =

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