Chapter 15 (2) Slope Stability 연세대학교 지반공학연구실

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil Assume : pore water pressure is zero is a trial circular arc that passes through the toe of the slope, and is the center of the circle. - Weight of soil wedge ABC = W = (Area of ABC) ( )

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil )

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil For equilibrium (Figure 15.22) 1. - resultant of the cohesive force (15.42) (15.43) 2. - the resultant of the normal and frictional forces along the surface of sliding

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil Friction circle : when or , the line of action of will make an angle of ’ with a normal to the arc and with thus be a tangent to a circle with and having . - The cohesion/unit area (Figure 15.22(c))

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil c’d based on a trial surface of sliding. The Max. Cohesion developed along the critical surface as (15.44) (15.45) where, = stability number in Figure 15.23 - For critical equilibrium - and or

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil  

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil  

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil  

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil Singh(1970) provided graphs of equal factors of safety for various slopes (Figure 15.25).  Example 15.6, 15.7

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil

15.9 Mass Procedure - Slope in Homogeneous c’ – Φ’ Soil

15.10 Ordinary Method of Slices (Fellenius 방법)

15.10 Ordinary Method of Slices (Fellenius 방법) th slice

15.10 Ordinary Method of Slices (Fellenius 방법) Equilibrium Consideration Resisting shear force (15.46) - Normal stress ’

15.10 Ordinary Method of Slices Equilibrium of the trial wedge (모멘트 평형), or (15.47) - error : 5 – 20 % (underestimate)  Example 15.8

15.10 Ordinary Method of Slices Note : The value of αn may be either positive or negative. The value of is positive when the slope of the arc is in the same quadrant on the ground slope.

15.11 Bishop’s Simplified Method of Slices Bishop(1955) proposed a more refined solution to the ordinary method of slices.

15.11 Bishop’s Simplified Method of Slices (15.49) Summing the forces in the vertical direction or (15.50)

15.11 Bishop’s Simplified Method of Slices Taking moment about (15.51) Where (15.52)

15.11 Bishop’s Simplified Method of Slices where, For simplicity, if we let (15.55)

15.11 Bishop’s Simplified Method of Slices Note : is present on both slices of Eq.(15.55). Hence, we must adopt a trial-and-error procedure to find the value of - Error in most cases ≤ 2% - Bishop’s simplified method is probably the most widely used.

15.12 Stability Analysis by Method of Slices for Steady State Seepage For steady state seepage through slopes, as is the situation in many practical cases, the pore water pressure must be considered when effective shear strength parameters are used. Eq.(14.58) for the ordinary method of slices will be modified (15.56)

15.12 Stability Analysis by Method of Slices for Steady State Seepage

15.12 Stability Analysis by Method of Slices for Steady State Seepage - Eq.(14.66) for Bishop’s method of slices will be modified (15.57) Note : is the total weight of the slice. Bishop and Morgenstern developed tables for calculation of for simple slopes. In Eq.(15.50), where average height of the th slice

15.12 Stability Analysis by Method of Slices for Steady State Seepage = 간극수압비 (15.59) Note is a nondimensional quantity (15.60) - For a steady state seepage case, (weighted average value) can be taken. may range up to 0.5

15.12 Stability Analysis by Method of Slices for Steady State Seepage (15.61) (15.62) where and = stability coefficients Table 14.1 gives the values of and for various combinations of , ’ and .

15.12 Stability Analysis by Method of Slices for Steady State Seepage - Determine from Table 15.3 step 1. obtain , and 2. obtain 3. From Table 14.1, obtain and for D=1, 1.25, and 1.5 4. Determine , using the values of m’ and n’ for each value of D 5. The required value of is the smallest one obtained in step 4.

15.15 Morgenstern ‘s Method of Slices for Rapid Drawdown Condition Morgenstern(1963) used Bishop’s method to determine the , during rapid drawdown.

15.15 Morgenstern ‘s Method of Slices for Rapid Drawdown Condition From Figure 15.41 L = Height of drawdown H = Height of embankment = Angle that the slope makes with the horizontal Assumption : 1. The embankment is made of homogeneous material, and rests on an impervious base. 2. Initially the water level coincides with the top of the embankment. 3. During drawdown, pore water pressure does not dissipate. 4. Figure 15.42 to 15.44 provide the drawdown stability charts

15.15 Morgenstern ‘s Method of Slices for Rapid Drawdown Condition

15.15 Morgenstern ‘s Method of Slices for Rapid Drawdown Condition

15.15 Morgenstern ‘s Method of Slices for Rapid Drawdown Condition

15.15 Morgenstern ‘s Method of Slices for Rapid Drawdown Condition

15.15 Morgenstern ‘s Method of Slices for Rapid Drawdown Condition

15.15 Morgenstern ‘s Method of Slices for Rapid Drawdown Condition

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay (포화점토제방)

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay Before construction of the embankment, the pore pressure at P can be expressed as (15.63) of the embankment along the potential surface of sliding (15.64)

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay For simplicity, - embankment construction is rapid - no drainage occurs during the construction average shear strength of the clay will remain constant from to . will gradually increase (Consolidation) (drained shear strength)

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay The general nature of the variation of with time. - initially decreases with time - At the end of construction ( ), is minimum - Beyond this point, continues to increase with drainage up to

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay Cuts in Saturated Clay

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay

15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay