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Geotechnical Engineering-II [Lec #28: Finite Slope Stability Analysis]

Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show.
Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.

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Geotechnical Engineering-II [Lec #28: Finite Slope Stability Analysis]

  1. 1. 1 Geotechnical Engineering–II [CE-321] BSc Civil Engineering – 5th Semester by Dr. Muhammad Irfan Assistant Professor Civil Engg. Dept. – UET Lahore Email: mirfan1@msn.com Lecture Handouts: https://groups.google.com/d/forum/geotech-ii_2015session Lecture # 28 20-Dec-2017
  2. 2. 2 SLOPE STABILITY ANALYSIS Finite Slope (Swedish Slip Circle Method) Assumptions: 1. Material of the slope is homogeneous. 2. Soil is purely cohesive in nature i.e. f = 0. 3. Failure surface has a curved/circular or spoon like surface. 4. Shear strength of the soil is uniformly distributed along failure plane. (only possible if f = 0)
  3. 3. 3 SLOPE STABILITY ANALYSIS Swedish Slip Circle Method (Cohesive soils (f=0)) NSLC A B 𝐹𝑂𝑆 = 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘ (𝑀 𝑅) π·π‘–π‘ π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘” π‘€π‘œπ‘šπ‘’π‘›π‘‘ (𝑀 𝐷)
  4. 4. 4 NSLC A B 𝐹𝑂𝑆 = 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘ (𝑀 𝑅) π·π‘–π‘ π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘” π‘€π‘œπ‘šπ‘’π‘›π‘‘ (𝑀 𝐷) π·π‘–π‘ π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘” π‘€π‘œπ‘šπ‘’π‘›π‘‘ 𝑀 𝐷 = π‘Š βˆ™ π‘₯ tr = c + sn tan f For saturated clay under undrained loading; f=0  tr = c = su 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘ 𝑀 𝑅 = 𝜏 π‘Ÿ βˆ™ 𝐴𝐡 βˆ™ 𝑅 𝑀 𝑅 = 𝑐 βˆ™ (𝑅 βˆ™ πœƒ) βˆ™ 𝑅 𝑀 𝑅 = 𝑐 βˆ™ πœƒ βˆ™ 𝑅2 𝐹𝑂𝑆 = 𝑀 𝑅 𝑀 𝐷 𝐹𝑂𝑆 = 𝑐 βˆ™ πœƒ βˆ™ 𝑅2 π‘Š βˆ™ π‘₯ SLOPE STABILITY ANALYSIS Swedish Slip Circle Method (Cohesive soils (f=0)) Case-I: No Tension Crack W x q R π‘Š = (π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ 𝐴𝐡𝐢𝐴 Γ— 1) Γ— 𝛾 β†’ q in radians
  5. 5. 5 W x q R NSLC A B𝐹𝑂𝑆 = 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘ (𝑀 𝑅) π·π‘–π‘ π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘” π‘€π‘œπ‘šπ‘’π‘›π‘‘ (𝑀 𝐷) 𝑀 𝐷 = π‘Š βˆ™ π‘₯ tr = c + sn tan f For saturated clay under undrained loading; f=0  tr = c = su 𝑀 𝑅 = 𝑐 βˆ™ πœƒ2 βˆ™ 𝑅2 𝐹𝑂𝑆 = 𝑀 𝑅 𝑀 𝐷 𝐹𝑂𝑆 = 𝑐 βˆ™ πœƒ2 βˆ™ 𝑅2 π‘Š βˆ™ π‘₯ SLOPE STABILITY ANALYSIS Swedish Slip Circle Method (Cohesive soils (f=0)) Case-II: Development of Tension Crack β„Ž 𝑑 = 2𝑐 𝛾 πΎπ‘Ž q2 FOS will reduce after development of tension crack [∡ q2 < q] π·π‘–π‘ π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘” π‘€π‘œπ‘šπ‘’π‘›π‘‘ 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘
  6. 6. 6 W x R C A tr = c = su SLOPE STABILITY ANALYSIS Swedish Slip Circle Method (Cohesive soils (f=0)) Case-III: Tension Crack filled with water β„Ž 𝑑 = 2𝑐 𝛾 πΎπ‘Ž q2 𝛾wht PW h 2 3 β„Ž 𝑑 𝑃 π‘Š = 1 2 𝛾 𝑀 βˆ™ β„Ž 𝑑 2 𝑀 𝐷 = π‘Š βˆ™ π‘₯ + 𝑀 𝑅 = 𝑐 βˆ™ πœƒ2 βˆ™ 𝑅2 𝐹𝑂𝑆 = 𝑐 βˆ™ πœƒ2 βˆ™ 𝑅2 π‘Š βˆ™ π‘₯ + 1 2 𝛾 𝑀 βˆ™ β„Ž 𝑑 2 β„Ž + 2 3 β„Ž 𝑑 FOS will reduce further when tension crack is filled with water 𝐹𝑂𝑆 = 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘ (𝑀 𝑅) π·π‘–π‘ π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘” π‘€π‘œπ‘šπ‘’π‘›π‘‘ (𝑀 𝐷) π·π‘–π‘ π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘” π‘€π‘œπ‘šπ‘’π‘›π‘‘ 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘ 1 2 𝛾 𝑀 βˆ™ β„Ž 𝑑 2 β„Ž + 2 3 β„Ž 𝑑
  7. 7. 7 Practice Problem #3 Determine the factor of safety of the cohesive slope shown in the figure for the following two cases; A. No tension crack B. 2m deep tension crack filled with water (q1 = 38Β°) NSLC A B gb = 17.75 kN/m3 Cu above line AD = 21.5 kPa Cu below line AD = 33.5 kPa W 3.1m q1= 40Β° R q2= 35Β° D D’ 4m 2m Area of ABCA = 90m2
  8. 8. 8 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils)  For c-f soils, normal stress would change along slip circle  Different normal stress means, shear resistance would also be different (∡ M-C equation)  Failing slope divided into slices TR Guidelines for Slice Selection  Slices do not have to be of equal width  For convenience, base arc of each slice should pass through one soil type only  Slice width should be limited (curved base approximated as straight line)
  9. 9. 9 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils) Wt a a TR l b 𝑙 = 𝑏 cos 𝛼 𝑙 = 𝑏 sec 𝛼 𝐹𝑂𝑆 = 𝑀 𝑅 𝑀 𝐷 TR = Total shear resistance force acting on slice 𝑇𝑅 = 𝜏 𝑅 Γ— (𝑙 βˆ™ 1) 𝑇𝑅 = (𝑐′ + 𝜎 𝑛′ tan πœ™) Γ— 𝑙 TR 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘ tR = Shear resistance (stress) offered by soil 𝑀 𝑅 = 𝑇𝑅 Γ— π‘Ÿ … (πΈπ‘ž. 1)
  10. 10. 10 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils) where, 𝜎 𝑛′ = π‘Šπ‘‘ cos 𝛼 𝑙 Γ— 1 𝑇𝑅 = 𝑐′ 𝑙 + π‘Šπ‘‘ cos 𝛼 𝑙 Γ— 1 𝑙 tan πœ™ 𝑇𝑅 = 𝑐′ 𝑙 + π‘Šπ‘‘ cos 𝛼 tan πœ™ 𝑀 𝑅 = π‘Ÿ (𝑐′ 𝑏 sec 𝛼 + π‘Šπ‘‘ cos 𝛼 tan πœ™) Wt a a a TR l b 𝑙 = 𝑏 cos 𝛼 𝑙 = 𝑏 sec 𝛼 Wt cos a Wt sin a 𝑀 𝑅 = 𝑇𝑅 Γ— π‘Ÿ … (πΈπ‘ž. 1) 𝑇𝑅 = 𝜏 𝑅 Γ— (𝑙 βˆ™ 1) 𝑇𝑅 = (𝑐′ + 𝜎 𝑛′ tan πœ™) Γ— 𝑙 𝑅𝑒𝑠𝑖𝑠𝑑𝑖𝑛𝑔 π‘€π‘œπ‘šπ‘’π‘›π‘‘ πΈπ‘ž. 1
  11. 11. 11 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils) Wt a a a TR l b 𝑙 = 𝑏 cos 𝛼 𝑙 = 𝑏 sec 𝛼 Wt cos a Components of disturbing force (Wt) 1. Wt cos a οƒ  Passes through center of rotation, i.e. zero moment 2. Wt sin a οƒ  Tangential component; causing sliding 𝑀 𝐷 = π‘Ÿ (π‘Šπ‘‘ sin 𝛼) Wt sin a π·π‘–π‘ π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘” π‘€π‘œπ‘šπ‘’π‘›π‘‘ TR
  12. 12. 12 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils) Wt a a a TR l b 𝑙 = 𝑏 cos 𝛼 𝑙 = 𝑏 sec 𝛼 Wt cos a 𝐹𝑂𝑆 = (𝑐′ 𝑏 sec 𝛼 + π‘Šπ‘‘ cos 𝛼 tan πœ™) (π‘Šπ‘‘ sin 𝛼) Wt sin a 𝐹𝑂𝑆 = 𝑀 𝑅 𝑀 𝐷 𝑀 𝑅 = π‘Ÿ (𝑐′ 𝑏 sec 𝛼 + π‘Šπ‘‘ cos 𝛼 tan πœ™) 𝑀 𝐷 = π‘Ÿ (π‘Šπ‘‘ sin 𝛼)
  13. 13. 13 CRITICAL SLIP CIRCLE NSL A B Many slip circles are possible on any slope Slip circle having minimum FOS οƒ  Critical Slip Circle / Critical Failure Plane Design has to satisfy safety against critical slip circle FOS = 2.4 1.20 1.79 1.55
  14. 14. 14 LOCATION OF CRITICAL SLIP CIRCLE In Cohesive soils (f=0) NSL A B  Plot the configuration according to scale  Draw two lines from point A and B at angles as has been shown in figure.  Taking radius equal to OA, draw a circle passing through the slope Case-I: Toe Failure (Fellenius Method) r q1 q2 Slope Slope Angle q1 q2 1V : 0.5H 60° 29° 40° 1 : 1 45° 28° 38° 1 : 1.5 34° 26° 35° 1 : 2 27° 25° 35° 1 : 3 19° 25° 35° Empirical values of q1 and q2
  15. 15. 15 LOCATION OF CRITICAL SLIP CIRCLE In Cohesive soils (f=0) NSL A B  Plot the configuration according to scale  Draw a vertical line at the mid-point of slope (the center of critical slip circle always lies on a vertical line passing through the mid-point of slope)  Determine the center by hit and trial method by comparing the FOS  Circle with minimum FOS is the critical circle Case-II: Base Failure (Fellenius Method) 𝐻 2 𝐻 2 FOS = 1.65 FOS = 1.10 133.5°  Angle made by critical circle at the center is about 133.5°. (Fellenius)
  16. 16. 16 NSL A B H H  Plot the configuration according to scale  Find the intersection point of β€˜4.5H’ horizontal and β€˜H’ distance vertical downward from A  Draw the direction angle q1 and q2  Join the points of intersection O and C  Locate OC by hit and trial. For this try O1, O2,…… and make circles.  The circle giving minimum FOS is the critical circle q2 q1 O1 O2 O3 Omin = Ocr LOCATION OF CRITICAL SLIP CIRCLE In c-f soils O4 4.5H C
  17. 17. 17 SHORT TERM AND LONG TERM STABILITY Clay Core Construction of dam core - Clay material - Very low permeability - Construction in layers with compaction at OMC SHORT TERM STABILITY Stability of slope immediately after construction  Undrained conditions  Undrained parameters (cu and fu) to be used for slope stability analysis  Obtained from UU or CU triaxial tests  Total unit weight of soil (gb) to be used  Called as TOTAL STRESS ANALYSIS  Change in pore water pressure totally dependent upon stress change
  18. 18. 18 SHORT TERM AND LONG TERM STABILITY Clay Core Construction of dam core - Clay material - Very low permeability - Construction in layers with compaction at OMC LONG TERM STABILITY Stability of slope long time after construction  Drained conditions  Drained parameters (cd (or c’) and fd (f’)) to be used for slope stability analysis  Obtained from CD triaxial tests, or CU tests with PWP measurements  Effective unit weight of soil (gsub (or g’)) to be used  Called as EFFECTIVE STRESS ANALYSIS  Change in pore water pressure independent of stress change
  19. 19. 19 THE END REFERENCE MATERIAL Principles of Geotechnical Engineering – (7th Edition) Braja M. Das Chapter #13 Essentials of Soil Mechanics and Foundations (7th Edition) David F. McCarthy Chapter #17 Geotechnical Engineering – Principles and Practices – (2nd Edition) Coduto, Yueng, and Kitch Chapter #17 In fact, this is just the beginning!

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