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

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

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

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

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 0.30-0.31 Steel 0.27-0.30 Cast iron 0.21-0.26 Concrete 0.20 Glass 0.18-0.3 Foam 0.10 to 0.40 Cork ~ 0.00 Auxetics negative soil   poisson's ratio   saturated clay 0.40-0.50 part. sat. clay 0.30- 0.40 dense sand 0.30-0.40 loose sand 0.10-0.30 granite 0.23-0.27 ell e/2 e/2 If v = 0.5, “incompressible” (no net volume change)

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

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

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

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

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

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

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

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

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

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

Headloss (flow) from B to D Example with Seepage Conditions Piezometer @ 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

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)

Piezometer @ 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’

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

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?

Quick Sand See the video: (G. Winters) See the video: http://dsc.discovery.com/videos/smash-lab-fluidized-sand.html

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 = 0 so no flow B 5’ A What is stress at B? Datum g = 110 pcf

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

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

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

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)

(Cedergren, 1989)

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