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

1. 1. 1 Geotechnical EngineeringβII [CE-321] BSc Civil Engineering β 5th Semester by Dr. Muhammad Irfan Assistant Professor Civil Engg. Dept. β 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 πΉππ = πππ ππ π‘πππ ππππππ‘ (π π) π·ππ π‘π’πππππ ππππππ‘ (π π·)
4. 4. 4 NSLC A B πΉππ = πππ ππ π‘πππ ππππππ‘ (π π) π·ππ π‘π’πππππ ππππππ‘ (π π·) π·ππ π‘π’πππππ ππππππ‘ π π· = π β π₯ tr = c + sn tan f For saturated clay under undrained loading; f=0 ο tr = c = su πππ ππ π‘πππ ππππππ‘ π π = π π β π΄π΅ β π π π = π β (π β π) β π π π = π β π β π2 πΉππ = π π π π· πΉππ = π β π β π2 π β π₯ SLOPE STABILITY ANALYSIS Swedish Slip Circle Method (Cohesive soils (f=0)) Case-I: No Tension Crack W x q R π = (π΄πππ ππ π΄π΅πΆπ΄ Γ 1) Γ πΎ β q in radians
5. 5. 5 W x q R NSLC A BπΉππ = πππ ππ π‘πππ ππππππ‘ (π π) π·ππ π‘π’πππππ ππππππ‘ (π π·) π π· = π β π₯ tr = c + sn tan f For saturated clay under undrained loading; f=0 ο tr = c = su π π = π β π2 β π2 πΉππ = π π π π· πΉππ = π β π2 β π2 π β π₯ SLOPE STABILITY ANALYSIS Swedish Slip Circle Method (Cohesive soils (f=0)) Case-II: Development of Tension Crack β π‘ = 2π πΎ πΎπ q2 FOS will reduce after development of tension crack [β΅ q2 < q] π·ππ π‘π’πππππ ππππππ‘ πππ ππ π‘πππ ππππ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 π‘ = 2π πΎ πΎπ q2 πΎwht PW h 2 3 β π‘ π π = 1 2 πΎ π€ β β π‘ 2 π π· = π β π₯ + π π = π β π2 β π2 πΉππ = π β π2 β π2 π β π₯ + 1 2 πΎ π€ β β π‘ 2 β + 2 3 β π‘ FOS will reduce further when tension crack is filled with water πΉππ = πππ ππ π‘πππ ππππππ‘ (π π) π·ππ π‘π’πππππ ππππππ‘ (π π·) π·ππ π‘π’πππππ ππππππ‘ πππ ππ π‘πππ ππππππ‘ 1 2 πΎ π€ β β π‘ 2 β + 2 3 β π‘
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 Dβ 4m 2m Area of ABCA = 90m2
8. 8. 8 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils) ο± For c-f soils, normal stress would change along slip circle ο± Different normal stress means, shear resistance would also be different (β΅ M-C equation) ο± Failing slope divided into slices TR Guidelines for Slice Selection ο± Slices do not have to be of equal width ο± For convenience, base arc of each slice should pass through one soil type only ο± 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 π = π cos πΌ π = π sec πΌ πΉππ = π π π π· TR = Total shear resistance force acting on slice ππ = π π Γ (π β 1) ππ = (πβ² + π πβ² tan π) Γ π TR πππ ππ π‘πππ ππππππ‘ tR = Shear resistance (stress) offered by soil π π = ππ Γ π β¦ (πΈπ. 1)
10. 10. 10 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils) where, π πβ² = ππ‘ cos πΌ π Γ 1 ππ = πβ² π + ππ‘ cos πΌ π Γ 1 π tan π ππ = πβ² π + ππ‘ cos πΌ tan π π π = π (πβ² π sec πΌ + ππ‘ cos πΌ tan π) Wt a a a TR l b π = π cos πΌ π = π sec πΌ Wt cos a Wt sin a π π = ππ Γ π β¦ (πΈπ. 1) ππ = π π Γ (π β 1) ππ = (πβ² + π πβ² tan π) Γ π πππ ππ π‘πππ ππππππ‘ πΈπ. 1
11. 11. 11 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils) Wt a a a TR l b π = π cos πΌ π = π sec πΌ Wt cos a Components of disturbing force (Wt) 1. Wt cos a ο  Passes through center of rotation, i.e. zero moment 2. Wt sin a ο  Tangential component; causing sliding π π· = π (ππ‘ sin πΌ) Wt sin a π·ππ π‘π’πππππ ππππππ‘ TR
12. 12. 12 SLOPE STABILITY ANALYSIS Ordinary Method of Slices (OMS) (c-f soils) Wt a a a TR l b π = π cos πΌ π = π sec πΌ Wt cos a πΉππ = (πβ² π sec πΌ + ππ‘ cos πΌ tan π) (ππ‘ sin πΌ) Wt sin a πΉππ = π π π π· π π = π (πβ² π sec πΌ + ππ‘ cos πΌ tan π) π π· = π (ππ‘ sin πΌ)
13. 13. 13 CRITICAL SLIP CIRCLE NSL A B ο±Many slip circles are possible on any slope ο±Slip circle having minimum FOS ο  Critical Slip Circle / Critical Failure Plane ο±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 ο± Plot the configuration according to scale ο± Draw two lines from point A and B at angles as has been shown in figure. ο± 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 60Β° 29Β° 40Β° 1 : 1 45Β° 28Β° 38Β° 1 : 1.5 34Β° 26Β° 35Β° 1 : 2 27Β° 25Β° 35Β° 1 : 3 19Β° 25Β° 35Β° Empirical values of q1 and q2
15. 15. 15 LOCATION OF CRITICAL SLIP CIRCLE In Cohesive soils (f=0) NSL A B ο± Plot the configuration according to scale ο± 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) ο± Determine the center by hit and trial method by comparing the FOS ο± Circle with minimum FOS is the critical circle Case-II: Base Failure (Fellenius Method) π» 2 π» 2 FOS = 1.65 FOS = 1.10 133.5Β° ο± Angle made by critical circle at the center is about 133.5Β°. (Fellenius)
16. 16. 16 NSL A B H H ο± Plot the configuration according to scale ο± Find the intersection point of β4.5Hβ horizontal and βHβ distance vertical downward from A ο± Draw the direction angle q1 and q2 ο± Join the points of intersection O and C ο± Locate OC by hit and trial. For this try O1, O2,β¦β¦ and make circles. ο± 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 ο± Undrained conditions ο± Undrained parameters (cu and fu) to be used for slope stability analysis ο± Obtained from UU or CU triaxial tests ο± Total unit weight of soil (gb) to be used ο± Called as TOTAL STRESS ANALYSIS ο± 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 ο± Drained conditions ο± Drained parameters (cd (or cβ) and fd (fβ)) to be used for slope stability analysis ο± Obtained from CD triaxial tests, or CU tests with PWP measurements ο± Effective unit weight of soil (gsub (or gβ)) to be used ο± Called as EFFECTIVE STRESS ANALYSIS ο± Change in pore water pressure independent of stress change
19. 19. 19 THE END REFERENCE MATERIAL Principles of Geotechnical Engineering β (7th Edition) Braja M. Das Chapter #13 Essentials of Soil Mechanics and Foundations (7th Edition) David F. McCarthy Chapter #17 Geotechnical Engineering β Principles and Practices β (2nd Edition) Coduto, Yueng, and Kitch Chapter #17 In fact, this is just the beginning!