Practice Problems: Slope Stability
Question 1
A geotechnical engineer is analyzing a 35-foot high slope in a homogeneous clay soil. The slope angle is 3H:1V (horizontal to vertical). The soil has a unit weight of 115 pcf, cohesion of 850 psf, and a friction angle of 0°. Using the infinite slope method, what is the factor of safety against slope failure?
(a) 1.95
(b) 2.18
(c) 2.45
(d) 2.67
Question 2
A civil engineer is evaluating a slope failure at a highway embankment. A circular slip surface has been identified with a radius of 45 meters and center coordinates (0, 40) m. The slope contains saturated clay with cohesion c = 25 kPa, φ = 0°, and γsat = 19.5 kN/m³. The length of the slip arc is 38 m. Using the φ = 0 method, what is the factor of safety per meter width?
(a) 1.12
(b) 1.28
(c) 1.45
(d) 1.63
Question 3
A geotechnical consultant is designing a stabilization system for a 25-meter high slope at 2H:1V in a c-φ soil. Soil properties: c' = 15 kPa, φ' = 28°, γ = 18.5 kN/m³. Analysis shows the critical failure surface requires a resisting force of 385 kN/m. What length of soil nail (capacity = 125 kN/m per 5 m length) is required?
(a) 12 m
(b) 15 m
(c) 18 m
(d) 20 m
Question 4
A slope stability analysis is being performed on a 42-foot high embankment with a 2.5H:1V slope. The soil has two layers: top layer (20 ft thick) with γ = 110 pcf, c = 400 psf, φ = 22°; bottom layer with γ = 125 pcf, c = 650 psf, φ = 18°. Using simplified Bishop method, the calculated mobilized shear strength equals 28,500 lb/ft. If the driving force is 22,800 lb/ft, what is the factor of safety?
(a) 1.15
(b) 1.25
(c) 1.35
(d) 1.45
Question 5
An engineer is analyzing seismic stability of a slope using the pseudostatic method. The slope is 18 m high at 2.5H:1V with soil parameters: γ = 19 kN/m³, c' = 20 kPa, φ' = 30°. The horizontal seismic coefficient kh = 0.15. Static factor of safety is 1.65. What is the approximate seismic factor of safety?
(a) 0.98
(b) 1.08
(c) 1.18
(d) 1.28
Question 6
A geotechnical engineer evaluates a slope with a planar failure surface. The slope height is 30 feet at 2H:1V. Soil properties: γ = 118 pcf, c' = 500 psf, φ' = 25°, pore pressure ratio ru = 0.35. The failure plane is parallel to slope surface at depth 12 feet. What is the factor of safety using infinite slope analysis?
(a) 1.32
(b) 1.48
(c) 1.64
(d) 1.82
Question 7
A mining engineer analyzes a rock slope with three joint sets. The critical joint set dips at 35° toward the excavation face (dip direction = 180°), while the slope face dips at 55° in the same direction. Joint properties: c = 50 kPa, φ = 32°, unit weight of rock = 26 kN/m³. For a 15 m high slope, what is the planar failure factor of safety?
(a) 1.45
(b) 1.62
(c) 1.78
(d) 1.95
Question 8
A coastal engineer is analyzing a slope above a beach where wave action has eroded the toe. Original slope was 3H:1V, height 22 m. Erosion removed 8 m horizontally at the toe, steepening the lower 12 m to 1.5H:1V. Soil: γ = 17.5 kN/m³, c' = 18 kPa, φ' = 26°. The factor of safety before erosion was 1.55. What is the approximate new factor of safety?
(a) 1.08
(b) 1.18
(c) 1.28
(d) 1.38
Question 9
A geotechnical engineer performs a back-analysis of a failed slope to determine residual strength parameters. The slope failed at 2.2H:1V, height = 35 feet, with water table at ground surface. Total unit weight = 122 pcf, saturated unit weight = 128 pcf. Analysis shows φr = 12°. Assuming FS = 1.0 at failure and cr = 0, what was the driving moment per foot of width?
(a) 142,000 lb-ft/ft
(b) 158,000 lb-ft/ft
(c) 174,000 lb-ft/ft
(d) 190,000 lb-ft/ft
Question 10
An engineer evaluates a remediation scheme for a slope with FS = 0.95. The slope is 28 m long, 16 m high at 2H:1V. Option A: Install drainage to lower water table 6 m, increasing FS by 0.35. Option B: Flatten slope to 3H:1V, increasing FS by 0.45 but requiring excavation of 850 m³/m. If excavation costs $28/m³ and drainage costs $185,000, which option is more economical?
(a) Option A by $38,000
(b) Option B by $38,000
(c) Option A by $52,000
(d) Option B by $52,000
Question 11
A transportation engineer analyzes a cut slope for a highway project. The slope is 45 feet high at 1.5H:1V in stiff clay. Parameters: γ = 115 pcf, cu = 1,250 psf, φu = 0°. Using Taylor's stability charts for φ = 0°, the stability number Ns = 5.52 for this slope angle and depth factor. What is the factor of safety?
(a) 1.38
(b) 1.52
(c) 1.66
(d) 1.80
Question 12
A dam engineer evaluates rapid drawdown conditions for a reservoir embankment. The upstream slope is 3H:1V, height 25 m. Soil properties: γsat = 20 kN/m³, c' = 15 kPa, φ' = 30°. Initial FS with full reservoir = 1.75. After rapid drawdown, the phreatic surface lags, creating ru = 0.45 in the slope. What is the approximate FS during drawdown?
(a) 0.98
(b) 1.08
(c) 1.18
(d) 1.28
Question 13
A forensic engineer investigates a landslide in residual soil. The failure surface is circular with radius 52 feet, arc length 68 feet. Back-calculation shows: total weight of sliding mass = 385,000 lb/ft, moment arm = 28 feet. Laboratory tests on failure surface samples indicate c' = 0 and φ'residual needs determination. Assuming FS = 1.0 at failure, what is φ'residual?
(a) 11.2°
(b) 13.8°
(c) 16.4°
(d) 19.0°
Question 14
A site engineer monitors a slope with inclinometers showing movement. The slope is 20 m high, 2.5H:1V, in clayey soil: γ = 18.8 kN/m³, c' = 22 kPa, φ' = 24°. Current FS = 1.15. To achieve FS = 1.50, the engineer proposes installing tiebacks at 3 m spacing (horizontal). What force per tieback is required if slope length is 45 m?
(a) 285 kN
(b) 340 kN
(c) 395 kN
(d) 450 kN
Question 15
A railroad engineer analyzes a slope adjacent to tracks where a 15-foot high cut will be made at 1.75H:1V. The soil profile shows: 0-8 ft: silty sand (γ = 108 pcf, c' = 0, φ' = 32°); 8-15 ft: sandy clay (γ = 122 pcf, c' = 650 psf, φ' = 22°). Water table is at 10 feet depth. What is the factor of safety using method of slices?
(a) 1.28
(b) 1.42
(c) 1.56
(d) 1.70
Question 16
A consulting engineer evaluates tension crack effects on slope stability. The slope is 32 m high at 2H:1V in heavily overconsolidated clay: γ = 19.2 kN/m³, cu = 85 kPa, φu = 0°. A tension crack 4.5 m deep has formed at the crest, filled with water to 3.8 m depth. What is the reduced factor of safety?
(a) 1.15
(b) 1.28
(c) 1.41
(d) 1.54
Question 17
A geotechnical engineer designs a reinforced slope using geosynthetics. The slope is 18 feet high at 1.5H:1V with soil parameters: γ = 125 pcf, c' = 200 psf, φ' = 28°. Analysis shows required reinforcement force = 2,850 lb/ft of slope width. Using geogrid with Tallow = 1,200 lb/ft and coverage ratio = 0.85, how many layers are needed?
(a) 2
(b) 3
(c) 4
(d) 5
Question 18
An engineer analyzes wedge failure in a rock slope. Two joint planes intersect: Joint 1 dips 42° at azimuth 165°, Joint 2 dips 38° at azimuth 235°. The slope face dips 65° at azimuth 185°. Rock unit weight = 25.5 kN/m³, joint cohesion = 35 kPa, joint friction = 35°. For a wedge volume of 185 m³, what is the factor of safety?
(a) 1.22
(b) 1.35
(c) 1.48
(d) 1.61
Question 19
A municipal engineer evaluates a landfill slope where waste and cover soil form a 40-foot high slope at 3H:1V. Waste properties: γ = 65 pcf, c' = 300 psf, φ' = 28°. A 2-foot thick clay cover exists on slope surface: γ = 118 pcf, c' = 450 psf, φ' = 18°. Critical failure surface extends through waste only. What is the factor of safety considering the cover surcharge?
(a) 1.38
(b) 1.52
(c) 1.66
(d) 1.80
Question 20
A geotechnical engineer performs probabilistic slope stability analysis. Mean values: H = 24 m, γ = 19 kN/m³, c' = 20 kPa (COV = 25%), φ' = 30° (COV = 12%), slope = 2.5H:1V. Deterministic FS = 1.42. Using first-order reliability method with calculated reliability index β = 2.15, what is the approximate probability of failure?
(a) 1.6%
(b) 2.8%
(c) 4.2%
(d) 5.5%
