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Unraveling Stress Distribution Mysteries: The Boussinesq Equation in Geotechnical Engineering

Explore the Boussinesq Equation to determine vertical and horizontal stresses under point loads in soil mechanics. Borrow and fill computations for proper soil compaction and understanding progressive slope failures. Calculate void ratio, moisture content, and more.

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Unraveling Stress Distribution Mysteries: The Boussinesq Equation in Geotechnical Engineering

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  1. Stress Distributionand all of its mysteries

  2. I. The Bulb of Pressure Force

  3. I. The Bulb of Pressure Force “what are the stresses at this point?”

  4. (boo-sinn-esk) 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”

  5. II. The Boussinesq EquationB. The Equation:

  6. II. The Boussinesq EquationB. The Equation: Where v = Poisson’s Ratio (0.48)

  7. 6.71

  8. If Poisson’s ratio is ~0.5, equations simplify to: σx = 3Px2z 2π R5 σy = 3Py2z 2π R5

  9. Your turn: Vertical load of 2500 lb/ft2 Determine horizontal and vertical stresses at X= 5, Y= 2, Z = 6

  10. Borrow & Fill Computations

  11. Cut and Fill Slopes FHWA, 2004 FHWA, 2004

  12. Proper OM & MD at time of compaction???

  13. ………………..Progressive Slope Failures Photo: Rick Wooten

  14. Borrow Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture Content = 15% Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry) Vv Vt

  15. Borrow Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture Content = 15% Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry) Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimalform)

  16. Borrow Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture Content = 15% Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry) Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimalform) Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15

  17. Borrow Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture Content = 15% Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry) Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimalform) Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15 Ws = 92 lbs/ft3

  18. Borrow Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture Content = 15% Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry) Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimalform) Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15 Ws = 92 lbs/ft3 Moisture Content = weight of water *100 weight of soil

  19. Borrow Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture Content = 15% Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry) Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimalform) Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15 Ws = 92 lbs/ft3 Moisture Content = weight of water *100 weight of dry soil 15 = Ww *100 92 lbs/ft3

  20. Borrow Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture Content = 15% Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry) Ws = Dry Density (lbs/ft3) = Wet Density (lbs/ft3) 1 + moisture content (decimalform) Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15 Ws = 92 lbs/ft3 Moisture Content = weight of water *100 weight of soil 15 = Ww *100 92 lbs/ft3 Ww = 14 lbs/ft3

  21. Specific Gravity (Gs) = 2.70 (unitless) Unit Weight water (γw) = 62.4 lbs per ft3

  22. Specific Gravity (Gs) = 2.70 (unitless) Unit Weight water (γw) = 62.4 lbs per ft3 Vol Soil (Vs) = Ws Gs * γw

  23. 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)

  24. 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

  25. 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

  26. 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…)

  27. 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

  28. 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

  29. Specific Gravity (Gs) = 2.70 (unitless) Unit Weight water (γw) = 62.4 lbs per ft3 Vv = 0.45 ft3 Vol Air = Va = Vv – Vw

  30. 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

  31. 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

  32. Specific Gravity (Gs) = 2.70 (unitless) Unit Weight water (γw) = 62.4 lbs per ft3 Vv = 0.45 ft3 Void Ratio (e) = Vv Vs

  33. Specific Gravity (Gs) = 2.70 (unitless) Unit Weight water (γw) = 62.4 lbs per ft3 Void Ratio (e) = Vv Vs = 0.45 0.55

  34. 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

  35. Fill Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture Content = 15% Specific Gravity = 2.7 Proctor Results: OM = 9% MD = 117 lbs/ft3 (dry) Vv Vt

  36. Fill Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture 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

  37. Fill Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture 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

  38. Fill Data: Wet density in situ = 106 lbs/ft3 Specific gravity = 2.70 Moisture 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

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