Stress Distribution and all of its mysteries

Stress Distributionand all of its mysteries

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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


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:



Va Air Wa

Vw Water Ww

Vs Soil Ws


Ws = Dry Density (lbs/ft3) = 106 lbs/ft3 1 + 0.15

Borrow Data:



Va Air Wa

Vw Water Ww

Vs Soil Ws 92



Ws = 92 lbs/ft3

Borrow Data:



Va Air Wa

Vw Water Ww

Vs Soil Ws 92

Moisture Content = weight of water *100 weight of soil



Ws = 92 lbs/ft3

Borrow Data:



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 = 92 lbs/ft3

Borrow Data:



Va Air Wa

Vw Water Ww 14

Vs Soil Ws 92


15 = Ww *100 92 lbs/ft3

Ww = 14 lbs/ft3



Ws = 92 lbs/ft3

Borrow Data:



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


Va Air Wa 0

Vw Water Ww 14

Vs Soil Ws 92


Vol Soil (Vs) = Ws Gs * γw


Va Air Wa 0

Vw Water Ww 14

Vs 0.55 Soil Ws 92


Vol Soil (Vs) = Ws = 92 lbs/ft3 Gs * γw (2.70)*(62.4 lbs)


Va Air Wa 0

Vw Water Ww 14

Vs 0.55 Soil Ws 92



Vs = 0.55 ft3


Va Air Wa 0

Vw Water Ww 14

Vs 0.55 Soil Ws 92



Vs = 0.55 ft3

Vol Voids (Vv) = Vt - Vs


Va Air Wa 0

Vw Water Ww 14

Vs 0.55 Soil Ws 92



Vs = 0.55 ft3

Vol Voids (Vv) = Vt - Vs Vv = 1 – 0.55

Vv = 0.45 ft3 (we will use this value later…)


Va Air Wa 0

Vw Water Ww 14

Vs 0.55 Soil Ws 92


Vv = 0.45 ft3

Vw = Ww = 14 lbs/ft3 γw 62.4 lbs


Va Air Wa 0

Vw 0.22Water Ww 14

Vs 0.55 Soil Ws 92


Vw = Ww = 14 lbs/ft3 γw 62.4 lbs

Vw = 0.22 ft3


Va Air Wa 0

Vw 0.22Water Ww 14

Vs 0.55 Soil Ws 92


Vv = 0.45 ft3

Vol Air = Va = Vv – Vw


Va Air Wa 0

Vw 0.22Water Ww 14

Vs 0.55 Soil Ws 92


Vv = 0.45 ft3

Vol Air = Va = Vv – Vw Va = 0.45 – 0.22


Va 0.23 Air Wa 0

Vw 0.22Water Ww 14

Vs 0.55 Soil Ws 92


Vol Air = Va = Vv – Vw Va = 0.45 – 0.22

Va = 0.23 ft3


Va 0.23 Air Wa 0

Vw 0.22Water Ww 14

Vs 0.55 Soil Ws 92


Vv = 0.45 ft3

Void Ratio (e) = Vv Vs


Va 0.23 Air Wa 0

Vw 0.22Water Ww 14

Vs 0.55 Soil Ws 92



= 0.45 0.55


Va 0.23 Air Wa 0

Vw 0.22Water Ww 14

Vs 0.55 Soil Ws 92



= 0.45 0.55

e = 0.82


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


Va Air Wa

Vw Water Ww

Vs Soil Ws

Fill Data:


Vv

Vt



Va Air Wa

Vw Water Ww

Vs Soil Ws

Fill Data:


Vv

Vt


9 = Ww *100 117 lbs/ft3


Va Air Wa

Vw Water Ww 10.5

Vs Soil Ws

Fill Data:


Vv

Vt


9 = Ww *100 117 lbs/ft3

Ww = 10.5 lbs/ft3


Va Air Wa

Vw Water Ww 10.5

Vs Soil Ws 117

Fill Data:


Vv

Vt

Ws = Dry Density (lbs/ft3) = MD = 117 lbs/ft3


Va Air Wa 0

Vw Water Ww 10.5

Vs Soil Ws 117

Fill Data:


Vv

Vt

Ws = Dry Density (lbs/ft3) = MD = 117 lbs/ft3

Fill Data:


Vv

Vt



Va Air Wa 0

Vw Water Ww 10.5

Vs Soil Ws 117

Fill Data:


Vv

Vt


Vs = 0.69 ft3 (compared to 0.55 from the borrow source)


Va Air Wa 0

Vw Water Ww 10.5

Vs 0.69 Soil Ws 117

Fill Data:


Vv

Vt


Va Air Wa 0

Vw Water Ww 10.5

Vs 0.69 Soil Ws 117

Vol Voids (Vv) = Vt - Vs

Fill Data:


Vv

Vt


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:


Vv

Vt


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:


Vv

Vt


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:


Volume of voids = 0.31

Vv

Vt


Va Air Wa 0


Vs 0.69 Soil Ws 117

Vol Air = Va = Vv – Vw Va = 0.31 – 0.17

Fill Data:


Vv

Vt


Va 0.14 Air Wa 0


Vs 0.69 Soil Ws 117

Vol Air = Va = Vv – Vw Va = 0.31 – 0.17

Va = 0.14 ft3

Fill Data:


Vv

Vt


Va 0.14 Air Wa 0


Vs 0.69 Soil Ws 117


Fill Data:


Vv

Vt


Va 0.14 Air Wa 0


Vs 0.69 Soil Ws 117


= 0.31

0.69

e = 0.45


Va 0.14 Air Wa 0


Vs 0.69 Soil Ws 117


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 + 0.8250,000 yd3 1 + 0.45

The Answer:



Volume of Borrow = 1 + 0.8250,000 yd3 1 + 0.45

Volume of Borrow = 62,759 yd3

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Stress Distribution and all of its mysteries