Theo van Asch, Utrecht University

Transcription

1 heo van Asch, Utrecht University

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4 Stresses on a plane σ3 δz σ σ τ b δx δs σ τ σ τ (σ σ sin σ sin σ (σ (σ ( b) - σcos (b) ( b) cos(b) - σ3sin( b) )sin( b) cos(b) + σ σ 3 3 ) + 2 (σ )sin(2b) σ 3 cos(b) )cos(2b)

5 he Mohr s envelope σ τ 2 2 (σ (σ + σ σ 3 3 ) + 2 (σ )sin(2b) σ 3 )cos(2b) σ3 σ 2b τ σ ind by means of the Mohr s circel an expression for τ,σ, the radius and the midpoint of the circle.

6 Morh s failure circle σ3 φ σ τ he strikeline on the circle gives a point here τ/σ has a maximum value So if failure occurs in a shear plane it must be that point : τ/σ equals then s/σ tanφ according to Coulomb (if there is no cohesion) σ

7 Morh s envelope c τ f τ 2f φ σ f σ 2f he strikeline along the failure circles connect the points here failure occurs. It is a line hich can be described by Coulombs la: τ f sc+σ f tanφ

8 Effective stress σ f effective stress!σ otal area X σ otal area grains Xm otal area ater X X σ X otal stress on plain X σ σ σ X σ X ( X m m m + σ + σ ) X ( X m ) σ σ X m + σ σ σ + σ

9 Effective stress Effective stress σ determines shear strength: X X s σ s c c σ - σ + σ tan φ σ - u + (σ - u) tan φ You can determine σ and u but not σ

10 otal stress and pore pressure otal stress σ is total eight above ABCD/area ABCD C A σ B he calculation of pore pressure u is a hydrological problem D

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12 Linear strain Deformation along vie line (x) x x 2 ε u x 2 2 u x u x δu δx u u 2 Constant linear strain l l δu δx u x u cons tan t x l l

13 Shear strain Deformation across vie line (y) y 2 u 2 u y 2 u y u y δu δy 2 tan y u x

14 Rheologic properties of material σ ε σstress εstrain σ σ Elastic σ ε ε σ t σ ε Viscous σ ε ε t σ ε Plastic σ0 ε t ε

15 Rheologic properties of material Elasto - plastic Elasto - viscous (Maxell) Elasto - viscous (Kelvin) Visco-plastic Bingham

16 Creep behaviour of soils Strain Creep failure Steady creep Initial elastic strain at σ,σ2,σ3 ime

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18 Principles of slope equilibrium W N Safety factor W N Strength of materials Mobilized shear strength Depends on material properties and equilibrium of forces in the ground (N ) Depends on equilibrium of forces in the ground ()

19 Principles of slope equilibrium W N W N U S C + (N - U) tanφ C + N tanφ

20 Principles of slope equilibrium W V Resolve forces perpendicular to shear plane: Wcos-N -U-Vsin0 Resolve forces parallel to shear plane: -Wsin-Vcos0 N U

21 Principles of slope equilibrium W V C + N tanφ N Wcos U Vsin Wsin + Vcos N U C + (Wcos U Vsin )tanφ Wsin + Vcos

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23 Infinite slope model E l Xl W E r N U X r he geometry of the slice is the same along the slope. X r and E r are equal to X l and E l. along the slope. So X and E can be cancelled in this equilibrium analysis or the stability calculation () resolve forces perpendicular and pararallel to shear surfaces

24 Infinite slope model E l Xl W E r N U X r Wcos N U 0 N Wsin 0 Wsin Wcos U C + C + N tanφ (Wcos U)tanφ Wsin

25 Principals of grounater flo and pore pressure calculation GW flo generated by difference in energy beteen points Energy expressed in m (head) Energy is sum of (ater) pressure head and elevation head (m) is called total head Equi-potential lines connect points ith same total head values Groundater flos perpendicular to equi-potential lines

26 Infinite slope model.principals of grounater flo and pore pressure calculation b Pressure head A 0 B x z γ u Equipotentiaal lijn Elevation head h 0 z γ s A otal head h x h B hz cos 2 xh is total head in B Elevation head in B 0 so pressure head in B h hz cos 2

27 Infinite slope model b z z γ u Equipotentiaal lijn γ s A h B b C c cos W (z z U z cos 2 )bγ γ u + z b cos bγ s hz cos 2 C + (Wcos U)tanφ Wsin

28 Infinite slope model b z z γ u Equipotentiaal lijn γ s A h B hz cos 2 c b cos + {[ ] 2 (z z )bγ z bγ cos z cos γ } b + u {(z z b)γ + z bγ }sin u s s cos tanφ

29 Infinite slope model { } { } s s U U U U 2 s s s u γ γ γ tan γ m(γ γ tanφ γ m(γ γ zcos c z z m tan tan γ γ γ zsin cos γ c z z tan tan zsin cos γ c 0 z tan tan 0 0,c z ϕ ϕ ϕ

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31 General equilibrium analyses: equilibrium of moment of forces d f he path e follo: R Expression for making use of safety factor definition El Xl P W ul Xr Er Expression for P by resolving forces vertically Expression for overall m, (hich includes P) by resolving moment forces

32 General equilibrium analyses: equilibrium of moment of forces d f R El W Xl Xr Er ul P s c S S { c + (σ u)tanφ } c l + (σ + (σ l - ul)tanφ σl P S u)tanφ c l l + (P ul)tanφ llength of slipsurface in slice n

33 General equilibrium analyses : equilibrium of moment of forces d f σl P S c l + (P ul)tanφ El Xl P W ul R Xr Er S c l + (P ul)tanφ { c l + (P ul)tanφ}

34 General equilibrium analyses : equilibrium of forces d f El Xl P ul W Xr Er R Vertical : P W (Xr m { c l + (P ul)tanφ} X ) cos + tan Pcos + sin l (c lsin tanφ W (X r P? X l) ultanϕ sin ) /m

35 General equilibrium analyses:equilibrium of moment of forces El Xl P ul W d f Xr Er llength of slipsurface in slice n R Wd X r m X l { c l + (P ul)tanφ} 0!and R { cl + (P ul)tanφ} + E Pf E (Wd Pf ) r l replace R m overall force equilibrium 0!

36 General equilibrium analyses :equilibriums of moment of forces d f R m { c l + (P ul)tanφ} Wd - Pf R El Xl P ul W Xr Er P W (X X ) r l (c lsin ultanφ sin ) /m Problem indeterminate : X r -X l 0 Bishop X r X l 0 Janbu X/Econstant Spencer X/Eλf(x) Morgenstern&Price

37 rom general equilibrium analyses to circular slip surface d f R m { cl + (P ul)tanφ} Wd - Pf R El Xl P ul W Xr Er Circular slipsurface f0 drsin Rconstant m { cl + (P ul)tanφ} Wsin

38 rom general Equilibrium analyses to circular slip surface ith Bishop assumption El Xl P ul W d f Xr Er R m { c l + (P ul)tanφ} P W (Xr m m Wsin X ) { c l + (W ul)tanφ }{ /m } l Wsin tanφ cos + tan (c lsin According to Bishop (X r -X l )0 for each individual slice ultanϕ sin ) /m