Stress Distribution and all of its mysteries
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Transcript of Stress Distribution and all of its mysteries
Stress Distributionand all of its mysteries
I. The Bulb of Pressure
Force
I. The Bulb of Pressure
“what are the stresses at thispoint?”
Force
II. The Boussinesq EquationA. Goal: to determine the vertical and horizontal stresses under a point load in a homogeneous, isotropic medium.“Allows us to determine vertical and horizontal stresses at any point in space”
(boo-sinn-esk)
II. The Boussinesq EquationB. The Equation:
II. The Boussinesq EquationB. The Equation:
Where v = Poisson’sRatio (0.48)
6.71
If Poisson’s ratio is ~0.5, equations simplify to:
σx = 3Px2z 2π R5
σy = 3Py2z 2π R5
Your turn:
Vertical load of 2500 lb/ft2Determine horizontal and vertical stresses at X= 5, Y= 2, Z = 6
Borrow & FillComputations
Cut and Fill Slopes
FHWA, 2004
FHWA, 2004
Proper OM & MD at time of compaction???
Photo: Rick Wooten
………………..Progressive Slope Failures
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws
Borrow Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws
Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimal form)
Borrow Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws
Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimal form)
Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15
Borrow Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws 92
Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimal form)
Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15
Ws = 92 lbs/ft3
Borrow Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws 92
Moisture Content = weight of water *100 weight of soil
Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimal form)
Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15
Ws = 92 lbs/ft3
Borrow Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws 92
Moisture Content = weight of water *100 weight of dry soil
15 = Ww *100 92 lbs/ft3
Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimal form)
Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15
Ws = 92 lbs/ft3
Borrow Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww 14
Vs Soil Ws 92
Moisture Content = weight of water *100 weight of soil
15 = Ww *100 92 lbs/ft3
Ww = 14 lbs/ft3
Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimal form)
Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15
Ws = 92 lbs/ft3
Borrow Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 14
Vs Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 14
Vs Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vol Soil (Vs) = Ws Gs * γw
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vol Soil (Vs) = Ws = 92 lbs/ft3 Gs * γw (2.70)*(62.4 lbs)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vol Soil (Vs) = Ws = 92 lbs/ft3 Gs * γw (2.70)*(62.4 lbs)
Vs = 0.55 ft3
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vol Soil (Vs) = Ws = 92 lbs/ft3 Gs * γw (2.70)*(62.4 lbs)
Vs = 0.55 ft3
Vol Voids (Vv) = Vt - Vs
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vol Soil (Vs) = Ws = 92 lbs/ft3 Gs * γw (2.70)*(62.4 lbs)
Vs = 0.55 ft3
Vol Voids (Vv) = Vt - Vs Vv = 1 – 0.55
Vv = 0.45 ft3 (we will use this value later…)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vv = 0.45 ft3
Vw = Ww = 14 lbs/ft3 γw 62.4 lbs
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw 0.22Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vw = Ww = 14 lbs/ft3 γw 62.4 lbs
Vw = 0.22 ft3
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw 0.22Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vv = 0.45 ft3
Vol Air = Va = Vv – Vw
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw 0.22Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vv = 0.45 ft3
Vol Air = Va = Vv – Vw Va = 0.45 – 0.22
Vol (ft3) Wt. (lbs/ft3)
Va 0.23 Air Wa 0
Vw 0.22Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vol Air = Va = Vv – Vw Va = 0.45 – 0.22
Va = 0.23 ft3
Vol (ft3) Wt. (lbs/ft3)
Va 0.23 Air Wa 0
Vw 0.22Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Vv = 0.45 ft3
Void Ratio (e) = Vv Vs
Vol (ft3) Wt. (lbs/ft3)
Va 0.23 Air Wa 0
Vw 0.22Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Void Ratio (e) = Vv Vs
= 0.45 0.55
Vol (ft3) Wt. (lbs/ft3)
Va 0.23 Air Wa 0
Vw 0.22Water Ww 14
Vs 0.55 Soil Ws 92
Specific Gravity (Gs) = 2.70 (unitless)Unit Weight water (γw) = 62.4 lbs per ft3
Void Ratio (e) = Vv Vs
= 0.45 0.55
e = 0.82
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Moisture Content = weight of water *100 weight of soil
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww
Vs Soil Ws
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Moisture Content = weight of water *100 weight of soil
9 = Ww *100 117 lbs/ft3
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww 10.5
Vs Soil Ws
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Moisture Content = weight of water *100 weight of soil
9 = Ww *100 117 lbs/ft3
Ww = 10.5 lbs/ft3
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa
Vw Water Ww 10.5
Vs Soil Ws 117
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Ws = Dry Density (lbs/ft3) = MD = 117 lbs/ft3
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 10.5
Vs Soil Ws 117
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Ws = Dry Density (lbs/ft3) = MD = 117 lbs/ft3
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol Soil (Vs) = Ws = 117 lbs/ft3 Gs * γw (2.70)*(62.4 lbs)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 10.5
Vs Soil Ws 117
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol Soil (Vs) = Ws = 117 lbs/ft3 Gs * γw (2.70)*(62.4 lbs)
Vs = 0.69 ft3 (compared to 0.55 from the borrow source)
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 10.5
Vs 0.69 Soil Ws 117
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 10.5
Vs 0.69 Soil Ws 117
Vol Voids (Vv) = Vt - Vs
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 10.5
Vs 0.69 Soil Ws 117
Vol Voids (Vv) = Vt - Vs Vv = 1 – 0.69
Vv = 0.31 ft3
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw Water Ww 10.5
Vs 0.69 Soil Ws 117
Vw = Ww = 10.5 lbs/ft3 γw 62.4 lbs
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw 0.17 Water Ww 10.5
Vs 0.69 Soil Ws 117
Vw = Ww = 10.5 lbs/ft3 γw 62.4 lbs
Vw = 0.17
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Volume of voids = 0.31
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va Air Wa 0
Vw 0.17 Water Ww 10.5
Vs 0.69 Soil Ws 117
Vol Air = Va = Vv – Vw Va = 0.31 – 0.17
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va 0.14 Air Wa 0
Vw 0.17 Water Ww 10.5
Vs 0.69 Soil Ws 117
Vol Air = Va = Vv – Vw Va = 0.31 – 0.17
Va = 0.14 ft3
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va 0.14 Air Wa 0
Vw 0.17 Water Ww 10.5
Vs 0.69 Soil Ws 117
Void Ratio (e) = Vv Vs
Fill Data:
Wet density in situ = 106 lbs/ft3Specific gravity = 2.70Moisture Content = 15%Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry)
Vv
Vt
Vol (ft3) Wt. (lbs/ft3)
Va 0.14 Air Wa 0
Vw 0.17 Water Ww 10.5
Vs 0.69 Soil Ws 117
Void Ratio (e) = Vv Vs
= 0.31
0.69
e = 0.45
Vol (ft3) Wt. (lbs/ft3)
Va 0.14 Air Wa 0
Vw 0.17 Water Ww 10.5
Vs 0.69 Soil Ws 117
Vol (ft3) Wt. (lbs/ft3)
Va 0.23 Air Wa 0
Vw 0.22Water Ww 14
Vs 0.55 Soil Ws 92
BORROW DATA FILL DATA
The Answer: How much borrow material is needed?
The Answer:
Volume of Borrow = 1 + eborrow
Volume of Fill 1 + efill
The Answer:
Volume of Borrow = 1 + eborrow
Volume of Fill 1 + efill
Volume of Borrow = 1 + 0.8250,000 yd3 1 + 0.45
The Answer:
Volume of Borrow = 1 + eborrow
Volume of Fill 1 + efill
Volume of Borrow = 1 + 0.8250,000 yd3 1 + 0.45
Volume of Borrow = 62,759 yd3