1. Total and Effective Stress: Hydrostatic and Flowing Conditions
GLE/CEE 330 Lecture Notes
Soil Mechanics
William J. Likos, Ph.D. Department of Civil and Environmental Engineering University of Wisconsin-Madison

2. Simplifying Assumptions
Soil is a continuum material
Soil is homogeneous for REV (Representative Elementary Volume)
Soil is isotropic (E, v)
Soil is linear-elastic (or others) sv sh t
(REV) E s e

3. Mechanics of Materials Review
Normal Stress, s
Shear Stress, t
Stress = Force/Area (e.g., psi, Pa)
Sign Convention:
compressive s is (+) for soils
counterclockwise t is (+) sz sx
tzx
(REV)
txz

4. Mechanics of Materials Review
Normal Stress-Strain
E = Young’s Modulus s e
Hooke’s Law s
s = Ee
e = DL/L DL L
Shear Stress-Strain
G = Shear Modulus t g t
g = shear strain

5. Mechanics of Materials Review
Poisson’s Ratio s
material  
poisson's ratio  
Rubber
~ 0.50
Magnesium
0.35
Titanium
0.34
Copper
0.33
Aluminium-alloy
Stainless steel
Steel
Cast iron
Concrete
0.20
Glass
Foam
0.10 to 0.40
Cork
~ 0.00
Auxetics
negative
soil  
poisson's ratio  
saturated clay
part. sat. clay
dense sand
loose sand
granite
ell
e/2
e/2
If v = 0.5, “incompressible”
(no net volume change)

6. Total Stress and Effective Stress
self-weight and external (induced) stress
stress carried by soil skeleton
s = total stress (sv and sh)
s’ = effective stress (s’v and s’h)
uw = pore water pressure (isotropic)
hydrostatic (no-flow) or flow cond.
EFFECTIVE STRESS GOVERNS SOIL
MECHANICAL BEHAVIOR (STRENGTH AND VOLUME CHANGE) P sv sh t uw

7. Vertical and Horizontal Stress
Vertical stress makes element want to expand laterally due to Poisson’s effect. However, it can’t because it is confined. This results in a horizontal stress that is typically less than the vertical stress.
Coefficient of Lateral Earth Pressure: y x z sz sx
For geostatic stress, x
and y are typically equal.
Special considerations: induced loads, slopes, retaining walls, tectonics

8. Total Vertical Stress, sv
Source: Self weight (geostatic stress) and stress from external loads
Geostatic Stress
Homogeneous Soil, g z W Fv

9. Total Vertical Stress, sv
Consider a layered soil profile
m= number of layers
gi = total unit weight of layer i
Hi = height (thickness) of layer i
g1, H1
g2, H2
g3, H3
g4, H4

10. Example sv (psf) Dry Sand 10’ g = 110 pcf A 1100 Sat Sand g = 120 pcf
2300 B z

11. Pore Pressure, uw Pore pressure is isotropic
Need to consider flow conditions: hydrostatic vs. flowing
Hydrostatic is simple; compute based on depth below water table (uw=0)
Need measurement for flow conditions (piezometer) or model with flow net
(Hydrostatic, No soil) zw
Define zw as (+) downward from W.T. A
(Hydrostatic, with soil)
piezometer
mudline zw zw zw A A A

12. Example uw or sv (psf) 10’ Water gw = 62.4 pcf A 624 sv Sat Sand
g = 120 pcf uw
10’ B
1248
2300 z

13. Effective Stress: s’ = s - uw
uw or sv or s’v (psf)
sv and s’v
50’
Sand
g = 110 pcf A
5500 z

14. What if the water table rises?
Rain
uw or sv or s’v (psf)
s’v
50’
s’v
Sand
g = 110 pcf A
2380
5500 z

15. Example uw or sv or s’v (psf) 20’ Dry Sand g = 110 pcf A 2200 30’ uw
Sat.Sand
g = 120 pcf
s’v B
1872
3928
5800 z

16. Headloss (flow) from B to D
Example with Seepage Conditions D
Head Diagram A
20’
Dry Sand
g = 118 pcf he B hp ht
20’ C
Sat.Sand
g = 125 pcf
10’ D
datum
Fractured Rock
10’
40’
Interpolating to C
So there is
Headloss (flow) from B to D

17. Dht=26’ Nd = 12.6 Dht/Nd = 26/12.6 = 2.1 htA = 94’–(3)(2.1) = 87.7’
heA = 43’
hpA = htA – heA = 44.8’
uA = 2796 psi
sA = ?
s’A = ?
(Lambe and Whitman)

18. D
Head Diagram A
20’
Dry Sand
g = 118 pcf he B hp ht
20’ C
Sat.Sand
g = 125 pcf
10’ D
datum
Fractured Rock
10’
40’

19. Example with Capillary Rise
(-)
(+)
uw or sv or s’v (psf) A 5’
Dry Silt, g = 100 pcf B
-624
500
1124
hc= 10’
g = 120 pcf uw C
1700
uw=0
20’ sv
g = 120 pcf
s’v D
1248
2852
4100

20. Implications: Precipitation-Induced Landslides
Rainfall
Factor of Safety
La Conchita California (2005)
10 confirmed fatalities
Potential failure plane
What happens if effective stress is reduced because pore pressure increases?

21. Quick Sand See the video:
(G. Winters)
See the video:

22. Seepage Pressure Dht = 0 so no flow What is stress at B? g = 110 pcf
Point
he (ft)
hp (ft)
ht (ft) A 25 B 5 20 C 15 10
10’ C
25’
10’
25’
Dht = so no flow B 5’ A
What is stress at B?
Datum
g = 110 pcf

23. so effective stress is reduced by upward flow
Seepage Pressure
Point
he (ft)
hp (ft)
ht (ft) A 35 B 5
26.7
31.7 C 15 10 25
10’ C
35’
10’
25’
so upward flow B 5’ A
What is stress at B?
Datum
g = 110 pcf
so effective stress is reduced by upward flow

24. Critical Hydraulic Gradient, icrit
What gradient for upward flow will cause quick condition? Dh g
“Buoyant Unit Weight” L A
Datum

25. Practical Implications: Piping and Critical Exit Gradients
flow
ht2 L
Consider an “element’ near
toe of the dam
ht1
If i > icrit, then unstable at toe
Toe

26. 1) Excessive gradient at toe removes material
2) This shortens length of flow path (increases gradient)
3) Causes piping to progress upstream and undermines dam
(Reddi, 2003)

27. (Cedergren, 1989)

28. Heave/Blowout May occur when s’ < 0 for clay piezometer
Sloped excavation
in clay, g = 100 pcf
30’
flow
10’ B
Artesian gravelly
Aquifer
Consider balance of vertical forces:
Area, A
10’ W
uBA
Net upward force may cause blowout of excavation floor