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## Slope Stability

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**Slope Stability**Crozet Tunnel 4281’**Gros Ventre Slide, WY, 1925**(pronounced “grow vahnt”) 50 million cubic yards**Earthquake Lake, MT, 1959**29 fatalities**Slope Stability**• Stresses and Strength • Applies to all sloping surfaces • Balancing of driving and resisting forces • If Resisting forces > Driving Forces: stability**Slope Stability**• Stresses and Strength • Applies to all sloping surfaces • Balancing of driving and resisting forces • If Resisting forces > Driving Forces: stability • Engineering Approach • Delineate the surface that is most at risk • Calculate the stresses • Calculate the Shear Strength**A Friendly Review From Last Month……**Stress on an inclined plane to Force σ = Force / Area Where is Normal Force and Shear Force = ?? cos Θ= a =Fn h = Fg sin Θ = o = Fs h=Fg Fn = Fg cos Θ Fs = Fg sin Θ**Shear Stress Analysis**Fn = Fg cos Θ Fs = Fg sin Θ Find the Shear stress What is behind this pretty little box???**Shear Stress Analysis**Fn = Fg cos Θ Fs = Fg sin Θ**Consider a planar slide whose failure surface is**‘linear’…..**II. Planar Slide—case 1**Volume of Slice = MO x PR x 0.5 x 1 ft**II. Planar Slide—case 1**Volume of Slice = MO x PR x 0.5 x 1 ft**Sa = Shear Stress**Sa = W sin β W = Fg Fn = Fg cos β Fs = Fg sin β**Sa = W sin β**Sr = Shear Resistance = (Friction + Cohesion)**Sa = W sin β**Fn = Fg cos β Fs = Fg sin β Sr = Friction + Cohesion = W cos β tan ϕ + c * (segment MO) Sr = W cos β tan ϕ + cL**Factor of Safety**Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**Sa = shear stress**Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!! Factor of Safety Fs = Sr = W cos β tan ϕ + cL Sa = W sin β**An Example…..**• Slope of 23 degrees • Angle of internal friction of 30 degrees • Cohesion of 90 lbs/ft2 • Soil is 100lbs/ft3 • MO has a distance of 100 ft • PR has a distance of 22 ft • Determine the factor of safety!!**III. Rotational Slide—case 1**• The process • Determine volume of each slice • Determine the weight of each slice**III. Rotational Slide—case 1**• The process • Determine volume of each slice • Determine the weight of each slice • Calculate the driving and resisting forces • of each slice • Sum ‘em up and let it rip!**III. Rotational Slide—case 1**“should use a minimum of 6 slices”**For Slice 1:**11’ x 20’ x 1’ = 220 ft3 220 ft3 x 100 lbs/ft3 = 22,000 lbs. For Slice 2: 30’ x 20’ x 1’ = 600 ft3 600 ft3 x 100 lbs/ft3 = 60,000 lbs Calculate the weight of each slice… For Slice 3: 38’ x 20’ x 1’ = 760 ft3 760 ft3 x 100 lbs/ft3 = 76,000 lbs For Slice 4: 25’ x 20’ x 1’ = 500 ft3 500 ft3 x 100 lbs/ft3 = 50,000 lbs**The Driving Force:**(+) (+)**The Driving Force:**(+) (+) (+) (+) (+) (-) (+) (-) (-) (-/+)**The Driving Force:**Your turn! 50,000 lbs 76,000 lbs 60,000 lbs 22,000 lbs