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Lecture 22 - SL16 Slope Stability by the Method of Slices o...

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SL16 Slope Stability by the Method of Slices o Allows for variability of c- φ slope soils (not homogenous) o Also allows for seepage and external forces o Slope is divided into a series of vertical slices which intersect a trial circular failure surface o The bottom of each slice is straight line and contains only one type of soil: o Draw free body diagram of a typical slice i and show the forces acting unit thickness: where: U i = Water pressure force T i = Shear force N i = Normal effective force E i = Normal forces on side S i = Shear force on sides W i = Weight of slice α i = Inclination of slice failure plane Δ x i = Width of slice Δ L i = Length of slice failure plane In general: 0 R M M Moment Driving Moment Resisting FS = = ( ) i i i i i i i tan ' c L L T φ σ + Δ = τ Δ = h L u L U and L N w i i i i i i i γ Δ = Δ = σ Δ = o Too many unknowns must make simplifying assumptions ..... 5 1 2 3 4 R 6 7 8 9 Soil 2, c 2 , φ 2 , γ 2 Soil 1, c 1 , φ 1 , γ 1 Soil 3, c 3 , φ 3 , γ 3 α α W i α i E i+1 S i+1 S i E i U i T i N i Δ L i Δ x i
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SL17 Geometry for method of slices: o for each slice must find weight, W i and base angle, α i : o can draw slope with slices to scale and measure with scale and protractor o or can use trigonometry: first define the circle center, the circle radius R, the width Δ x i of each slice, and the horizontal distance, x i , from the circle center to the slice center since = α α = R x sin then sin R x i 1 - i i i the volume of the slice is Δ x i h i where h i = height of the slice at its center and i i i y - cos R h Δ α = where Δ y i = distance to circle center above slice surface the weight of the slice, W i = γΔ x i h i Slice Geometry Example: toe circle, i = 45°, center above toe, 5' slices, γ = 120 pcf, R = 40', H = 30', Y = 40' for slice 3: x 3 = 12.5', Δ x 3 = 5' ° = = α 21 . 18 40' 12.5' sin 1 - 3 Δ y 3 = (40' - 12.5') = 27.5' 0.5' 1 ' 5 . 27 21 . 18 cos ' 40 y cos R h 3 3 3 = - ° = Δ - α = k 30 . 6 lb 1000 k 1 ) ' 5 . 10 )( ' 5 ( pcf 120 h x W 3 3 3 = = Δ γ = R α i α i x i = R sin α i h i H y i = R cos α i Δ y i Y Δ x i 2 4 5 1 3 6 7 8 H = 30' 45° x 3 y 3 h 3 Δ y 3 R = 40' Δ x 3 = 5'
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SL18 External Forces in Method of Slices: o method of slices very flexible: o weight of slice may include both saturated and moist soil o slices may include more than one type of soil o weight of slice should include any water above slice (i.e. calculate total slice weight, the water pressure will be subtracted from N i ) o add vertical external forces to slice weight o add horizontal external forces x moment arm to resisting or driving moment, as appropriate w i3 w i2 w i3 w i1 w iw w i2 R α i α i h w Δ x i P P H P V Q = q Δ x i r PH q
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SL19 Ordinary Method of Slices Fellinius (1927) o Assume resultants of side forces (E and S) are equal, opposite, and coincidental (therefore they cancel) o Resisting Moment: i i i i i i i i i L u cos W N and cos W U N Δ - α = α = + φ Δ - α + Δ = φ +
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