CE643_Spring2023_Week11_GS (1)

4/4/2023 1 CE 643 - Advanced Foundation Engineering Spring 2023 Week 11 04/04/2023 CONTENT – Slope Stability Analysis  Introduction  Failure of infinite slope  Rotational slide (circular failure)  Homogeneous soli slope: Design charts  Multi-layered soil slope: Method of slices 1 2

4/4/2023 2 Landslides  Landslides due to unstable natural or artificial slopes occur when the shear stresses exceed the shear strength .  The factors leading to instability can generally be classified as: a) Those causing increased stress • Increased unit weight of soil by wetting due to rainfall • Added external or surcharge loads such as buildings • Steepened slopes either by natural erosion or excavation of the toe • Applied cyclic loads due to earthquake b) Those causing reduction in strength • Absorption of water and subsequent swelling of double layer • Increased pore pressure • Cyclic loads • Loss of cementing materials • Weathering process • Strength loss with excessive strain of sensitive clays • Freezing and thawing  The presence of water in the slope causes both increase in stress and reduction in strength. From Professor D. T. Bergado’s notes (AIT) Major types of landslide movement Source: https://pubs.usgs.gov/fs/2004/3072/pdf/fs2004-3072.pdf 3 4

4/4/2023 4 Limit Equilibrium Methods -Examples From Professor Bruno Gonçalves da Silva’s notes (NJIT) Soil slope failure 1.Failure of infinite slope 2.Translation slide (plane failure) 3.Rotational slide (circular failure)  Homogeneous soli slope  Multi-layered soil slope 7 8

4/4/2023 5 Various Definition of Factor of Safety Source: dot.ny.gov Infinite slope failure in dry sand 9 10

4/4/2023 7 Special case (2): 𝐹𝑆 = 1, 𝑐 ᇱ ≠ 0 𝐹𝑆 = 𝑐 ᇱ + 𝛾 ௕ 𝑐𝑜𝑠 ଶ 𝛼 𝑧 𝑡𝑎𝑛∅ ᇱ 𝛾 ௕ 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 𝑧 = 1 𝑐 ᇱ + 𝛾 ௕ 𝑐𝑜𝑠 ଶ 𝛼 𝑧 𝑡𝑎𝑛∅ ᇱ = 𝛾 ௕ 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 𝑧 𝑐 ᇱ 𝑧𝛾 ௕ = 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 − 𝑐𝑜𝑠 ଶ 𝛼 𝑡𝑎𝑛∅ ᇱ 𝑐 ᇱ 𝑧𝛾 ௕ = 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼 𝑐𝑜𝑠𝛼 − 𝑐𝑜𝑠 ଶ 𝛼 𝑡𝑎𝑛∅ ᇱ 𝑐 ᇱ 𝑧𝛾 ௕ = 𝑐𝑜𝑠 ଶ 𝛼 𝑡𝑎𝑛𝛼 − 𝑐𝑜𝑠 ଶ 𝛼 𝑡𝑎𝑛∅ ᇱ 𝑐 ᇱ 𝑧𝛾 ௕ = 𝑐𝑜𝑠 ଶ 𝛼 (𝑡𝑎𝑛𝛼 − 𝑡𝑎𝑛∅ ᇱ ) Can be used to calculate the critical height 𝑧 = 𝐻 ௖௥ Example: 𝛾 ௕ = 60 𝑝𝑐𝑓 , ∅ ᇱ = 28° , 𝑐 ᇱ = 200 𝑝𝑠𝑓 , 𝛼 = 35° ,  If FS = 1, H ୡ୰ =?  If z = 20ft, FS =?  If z = 40ft, FS =? Infinite slope failure in soil with parallel seepage 13 14

4/4/2023 8 Infinite slope failure – Seepage parallel to slope 𝐵 𝑄 ௅ 𝑄 ோ 𝑆 ௠ 𝑃′ 𝐿 𝛼 𝛼 𝐿 𝜎 ே 𝜏 𝑧 D 𝑢 𝛾 ௧ 𝛾 ௗ 𝑊 = 𝛾 ௗ 𝐵𝐷 + 𝛾 ௧ 𝐵 𝑧 − 𝐷 = 𝐿 𝑐𝑜𝑠𝛼 𝛾 ௗ 𝐷 + 𝛾 ௧ 𝑧 − 𝐷 𝜎 ே = ௉ ௅ = ௐ௖௢௦ఈ ௅ = 𝛾 ௗ 𝐷 + 𝛾 ௧ 𝑧 − 𝐷 𝑐𝑜𝑠 ଶ 𝛼 𝜏 = ௌ ೘ ௅ = ௐ௦௜௡ ௅ = 𝛾 ௗ 𝐷 + 𝛾 ௧ 𝑧 − 𝐷 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 𝑢 = 𝛾 ௪ 𝑍 − 𝐷 𝑐𝑜𝑠𝛼 𝐿 𝑐𝑜𝑠𝛼 = L𝛾 ௪ 𝑍 − 𝐷 𝑐𝑜𝑠 ଶ 𝛼 𝜇 = 𝑢 𝐿 = 𝛾 ௪ 𝑍 − 𝐷 𝑐𝑜𝑠 ଶ 𝛼 𝜎 ே ᇱ = 𝜎 ே − 𝜇 = 𝛾 ௗ 𝐷 + (𝛾 ௧ −𝛾 ௪ ) 𝑧 − 𝐷 𝑐𝑜𝑠 ଶ 𝛼 𝛾 ௕ Failure criteria (Mohr-Coulomb criterion) 𝐹𝑆 = 𝜏 ௙ 𝜏 = 𝑐 ᇱ + 𝜎 ே ᇱ 𝑡𝑎𝑛∅ ᇱ 𝛾 ௗ 𝐷 + 𝛾 ௧ 𝑧 − 𝐷 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 𝐹𝑆 = 𝑐 ᇱ + 𝛾 ௗ 𝐷 + (𝛾 ௧ −𝛾 ௪ ) 𝑧 − 𝐷 𝑐𝑜𝑠 ଶ 𝛼 𝑡𝑎𝑛∅ ᇱ 𝛾 ௗ 𝐷 + 𝛾 ௧ 𝑧 − 𝐷 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 Special case (1): 𝑐 ᇱ = 0 𝐹𝑆 = 𝛾 ௗ 𝐷 + (𝛾 ௧ −𝛾 ௪ ) 𝑧 − 𝐷 𝑐𝑜𝑠 ଶ 𝛼 𝑡𝑎𝑛∅ ᇱ 𝛾 ௗ 𝐷 + 𝛾 ௧ 𝑧 − 𝐷 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 𝐹𝑆 = 𝛾 ௗ 𝐷 + (𝛾 ௧ −𝛾 ௪ ) 𝑧 − 𝐷 𝑡𝑎𝑛∅ ᇱ 𝛾 ௗ 𝐷 + 𝛾 ௧ 𝑧 − 𝐷 𝑡𝑎𝑛𝛼 When 𝐷 = 0, 𝐹𝑆 = (𝛾 ௧ −𝛾 ௪ )𝑡𝑎𝑛∅ ᇱ 𝛾 ௧ 𝑡𝑎𝑛𝛼 = 𝛾 ௕ 𝛾 ௧ 𝑡𝑎𝑛∅ ᇱ 𝑡𝑎𝑛𝛼 15 16

4/4/2023 10 Rotational Failure in Cohesive Soils ( = 0) Source: DM 7.01  For slopes in cohesive soils having approximately constant strength with depth to determine the factor of safety.  Undrain conditions  With surcharge, tension cracks, or submergence of slope, apply corrections of to determine safety factor. Rotational Failure in Cohesive Soils ( = 0) Source: DM 7.01 19 20

4/4/2023 11 Taylor’s (1948) Stability Charts Source: FHWA-NHI–06-088 Taylor’s (1948) Stability Charts (Cont.) 21 22

4/4/2023 13 Force acting on a typical slice Ordinary Method of Slices  There are several available methods that can be used to perform a circular arc stability analysis for an approach embankment over soft ground.  The simplest basic method is known as the Normal or Ordinary Method of Slices , also known as Fellenius’ method (Fellenius, 1936) or the Swedish circle method of analysis. Figure 6-7. Geometry of Ordinary Method of Slices Source: FHWA NHI-06-088 25 26

4/4/2023 14 Ordinary Method of Slices: Steps of calculations Figure 6-8. Example of dividing the failure mass in slices Source: FHWA NHI-06-088 Step 1. Draw a cross-section of the embankment and foundation soil profile on a scale of either 1-inch = 10 feet or 1-inch = 20 feet scale both horizontal and vertical. Step 2. Select a circular failure surface such as shown in Figure 6-7. Step 3. Divide the circular mass above the failure surface into 10 - 15 vertical slices as illustrated in Figure 6-8. Also, as shown in Figure 6-9 and 6-10 the driving and resisting forces of each slice act at the intersection of a vertical line drawn from the center of gravity of the slice to the failure circle to establish a centroid point on the circle. Figure 6-9 . Forces on a slice without water effect Figure 6-10. Forces on a slice with water effect Source: FHWA NHI-06-088 27 28

4/4/2023 16 Ordinary Method of Slices (Fellenius) Moment Equilibrium Horizontal Force Equilibrium 𝑰 𝑵 𝑰 𝑺 Method YES Ignore Ignore Ignore Ordinary Method of Slices YES Ignore YES Ignore Bishop Method (Bishop, 1955) Ignore YES YES Ignore Simplified Janbu Method (Janbu, 1954) YES YES YES YES Spencer Method (Spencer, 1967) Typical static forces on a slice of sliding mass without seepage. Different Stability Methods 31 32

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4/4/2023 19 EXAMPLE Example computation for ordinary method of slices. 37 38

4/4/2023 20 Example computation for Bishop's modified method. THANK YOU! 39 40