[Terzaghi] Unsaturated Soil Mechanics (2007) The effective stress principle, conventionally applied in saturated soils, is reviewed for constitutive modelling purposes. The assumptions for the applicability of Terzaghi’s single effective stress are recalled and its advantages are inventoried. The possible stress frameworks applicable to unsaturated soil modelling are reassessed in a comparative manner, specifically the Bishop’s single effective stress, the independent stress variables approach and the generalized stress framework. The latter considerations lead to the definition of a unified stress context, suitable for modelling soils under different saturation states. In order to qualify the implications brought by the proposed stress framework, several experimental data sets are re-examined in the light of the generalized effective stress. The critical state lines (CSLs) at different saturation states tend to converge remarkably towards a unique saturated line in the deviatoric stress versus mean effective stress plane. The effective stress interpretation is also applied to isotropic paths and compared with conventional net stress conception. The accent is finally laid on a second key feature for constitutive frameworks based on a unified stress, namely the sufficiency of a unique mechanical yield surface besides the unique CSL. Copyright q 2007 John Wiley & Sons, Ltd.
Unsaturated Soil Mechanics in Engineering Professor Delwyn G Fredlund University of Saskatchewan Saskatoon, SK, Canada GeoFrontiers 2005, Austin, Texas Geo-Institute, ASCE January 23-26, 2005 • Karl Terzaghi elevated Soil • • • • John Wiley & Sons, 1943 Mechanics from an Art to a Science Effective Stress, (σ – uw), for describing mechanical behavior of saturated soils Chapter 14 “Capillary Forces” (Also Chapter 15) Biot (1941) addressed consolidation of unsaturated soils Concepts from Agriculture (Baver, 1940) Unsaturated Soil Mechanics Problems Described in “Theoretical Soil Mechanics” by K Terzaghi (1943) Unsaturated Soil Mechanics in Engineering • • • • • • • • Introduction Challenges to Implementation Description of the Stress State Fundamental Constitutive Relations Role of the Soil-Water Characteristic Curve Use of SWCC in the Constitutive Relations Solution of a Series of PDEs Modeling Unsaturated Soils Problems Objectives • To illustrate the progression from theories and formulations to practical engineering protocols for solving a variety of unsaturated soil mechanics problems (e.g., seepage, shear strength and volume change), through use of “direct” and “indirect” characterization of unsaturated soil property functions • To describe the Challenges Faced and the Solutions Generated in moving towards the Implementation of Unsaturated Soil Mechanics Gradual Emergence of Unsaturated Soil Mechanics • 1950s: Independent measurement of pore-air and pore-water pressure through use of high air entry ceramic disks • 1960s: Laboratory testing of unsaturated soils • 1970s: Constitutive relations proposed and tested for uniqueness for unsaturated soils • 1980s: Solving formulations for classic Boundary Value Problems • 1990s: Establishing procedures for determination of unsaturated soil property functions • 2000+: Implementation into routine engineering practice Challenges to the Implementation of Unsaturated Soil Mechanics • Challenge #1: – To discover appropriate Stress State Variables for describing the physical behavior of unsaturated soils • Solution #1: ? Challenges to the Implementation of Unsaturated Soil Mechanics • Challenge #2: To develop devices that could measure a wide range of negative pore-water pressures (i.e., high matric suctions) • Solution #2: ? Challenges to the Implementation of Unsaturated Soil Mechanics • Challenge #3: – To develop (and test for uniqueness) constitutive relations suitable for describing unsaturated soil behavior • Solution #3: ? Challenges to the Implementation of Unsaturated Soil Mechanics • Challenge #4: – To overcome the excessive costs associated with the determination (i.e., measurement) of unsaturated soil properties (i.e., nonlinear functions) • Solution #4: ? Partial Differential Equation for SaturatedUnsaturated Water Flow Analysis Head variable to be solved w ∂ k ∂k ∂h ∂h y ∂h w ∂ h w ∂ h w + ky + + kx = − m2 γ w 2 ∂x ∂x ∂y ∂x ∂y ∂y ∂t w x Water coefficient of permeability (function of soil suction) Water storage (function of soil suction) Time Partial Differential Equation for Unsaturated Air Flow Analysis Pore-air pressure (primary variable to be solved) ∂ u a ∂k a ⎛ ∂u a ⎞ ∂k a ⎛ ∂u a ∂ 2ua ⎜⎜ + + ka ka ⎜ ⎟+ 2 ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ∂y ∂x Air coefficient of permeability (function of soil suction) ⎞ ⎛ e w ⎞ ω a g ∂u a ⎟⎟ = −⎜ S a − u a m2 ⎟ ⎝1+ e ⎠ RT ∂t ⎠ Air storage and compressibility (function of soil suction) Time Partial Differential Equation for SaturatedUnsaturated Stress-Deformation Analysis ∂ ⎡ ∂u ∂v ⎤ ∂ ⎡ ⎛ ∂u ∂v ⎞⎤ + ⎟⎟⎥ = + D12 ⎥ + ⎢ D44 ⎜⎜ ⎢ D11 ∂x ⎣ ∂x ∂y ⎦ ∂y ⎣ ⎝ ∂y ∂x ⎠⎦ X– Y– ⎛ ∂u ∂v ⎞ ⎤ ∂ ⎡ ∂ ⎡ ∂u ∂v ⎤ + D11 ⎥ + γ t = ⎢ D44 ⎜ + ⎟ ⎥ + ⎢ D12 ∂x ⎣ ∂x ∂y ⎦ ⎝ ∂y ∂x ⎠ ⎦ ∂y ⎣ D11, D12, D44 = Combination of E and µ which are function of soil suction and net total stresses Stress-deformation analyses have a degrees of freedom in each of the Cartesian coordinate directions Convergence of Nonlinear Partial Differential Equations • Convergence is the single most pressing problem facing modelers • Most successful solutions have involved Adaptive Grid Refinement methods, AGR (Oden, 1989; Yeh, 2000) • Mesh is dynamically upgraded during the solution based on error estimates • AGR becomes extremely important when solving the nonlinear PDEs associated with Unsaturated Soil Mechanics Two-dimensional seepage analysis through an earthfill dam with a clay core Optimized mesh for saturatedunsaturated seepage analysis Equipotential lines Problem illustrating the solution of a 3-dimensional, saturated-unsaturated seepage PDE Optimized, automatically generated finite element mesh Modeling of a waste tailings pond Stress analysis PDE combined with the Dynamic Programming procedure to compute the factor of safety DP Ge ne te d Critic a l S lip S urfa ce 30 FOS = 1.3 25 Shape and location of the slip surface are a part of the solution DP Search Bounda ry 20 15 10 Finite Ele me nt S he a r S tre ss 20 40 Dista nce 60 80 Prediction of Heave or Collapse of a Soil • Requires the solution of a saturated-unsaturated seepage model and a stress-deformation model Coupled Uncoupled Pseudo-coupled Saturated-Unsaturated Seepage Model Computes changes in matric suction Saturated-Unsaturated Stress-Deformation Model Computes deformations Scenario of Edge Lift for a Flexible Impervious Cover Boundary conditions and initial conditions must be specified both seepage and stress-deformation SVFlux Infiltration, q Flexible cover Depth, m Flux = Flux = CL Constant suction = 400 kPa 3 Distance from centre of cover or slab, m 12 SVFlux and SVSolid Can have one optimized Adaptive Mesh generated for seepage model and another for the stress-deformation model Depth, m Concrete slab C L 3 Distance from center, m 12 Matric Suction at Ground Surface after One Day of Infiltration for Various Infiltration Rates Matric suction, kPa Distance under slab 500 Initial 400 q = 10 mm/day 300 Specified zero suction q = 20 q = 30 200 q = 40 100 C L q = 50 q = 60 0 10 Distance from centre of cover, m 12 SVFlux Vertical Displacements at Ground Surface after One Day of Infiltration Distance under slab 25 Heave,(mm) specified zero suction 20 C L 15 q = 60 mm/day 10 q = 50 q = 40 q = 30 q = 20 q = 10 0 10 Distance from centre of cover, m 12 SVSolid Challenges to the Implementation of Unsaturated Soil Mechanics • Challenge #6: – To promote implementation of unsaturated soil mechanics into engineering practice • Solution #6: - Educational materials and visualization tools have been produced to better teach and understand unsaturated soil mechanics Concluding Remarks • Unsaturated Soil Mechanics needs to be first understood from the standpoint of the Constitutive equations describing soil behavior • Constitutive Equations can be written in terms of the SWCC for the soil which are then known as Unsaturated Soil Property Functions, USPF • Direct and Indirect procedures are available for the assessment of the SWCC • It is always possible to obtain an estimate of the required Unsaturated Soil Property Functions for geotechnical engineering applications Karl Terzaghi deserves credit not only for the fundamentals of saturated soil behavior but also for the fundamentals of unsaturated soil behavior Geo-Institute, Austin, Texas January 23-26, 2005 Thank You ... consolidation of unsaturated soils Concepts from Agriculture (Baver, 1940) Unsaturated Soil Mechanics Problems Described in “Theoretical Soil Mechanics by K Terzaghi (1943) Unsaturated Soil Mechanics. .. Unsaturated Soil Mechanics • Challenge #6: – To promote and teach unsaturated soil mechanics at universities and in engineering practice • Solution #6: ? Local vertical zones of unsaturated soils... the Solutions Generated in moving towards the Implementation of Unsaturated Soil Mechanics Gradual Emergence of Unsaturated Soil Mechanics • 1950s: Independent measurement of pore-air and pore-water