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Introduction to numerical modelling finite elements Linear elasticity Basic concepts of plasticity and Mohr Coulomb model Non linear finite elements and solution techniques Applied theory: Introduction to PLAXIS T01 Tutorial: Soil testing tool, MohrCoulomb Applied theory: Shallow foundations Applied theory: Structural elements and interfaces T02 Tutorial: Shallow foundation Drainedundrained analysis Consolidation analysis Applied theory: Soil parameters for drained and undrained analysis Applied theory: Slope stability and phic reduction T03 Tutorial: Consolidation and phic reduction Tentative schedule: Jan 1620, 2012 Critical state models Applied theory: Soil parameter for critical state models T04 Tutorial: Analysis of an embankment (inc. stability) Applied theory: Analysis of an embankment T05 Tutorial: Boston embankment (I) Hardening Soil Model and Small Strain Stiffness Applied theory: Soil parameters for Hardening Soil model Applied theory: Excavations Tentative schedule May 28May 21, 2012 T06 Tutorial: Excavation in Limburg Anisotropy, bonding and creep Applied theory: Numerical modelling of ground improvement T07 Tutorial: Boston embankment (II)
Numerical Methods of Geotechnics Prof Minna Karstunen University of Strathclyde Tentative schedule: Jan 16-20, 2012 Introduction to numerical modelling & finite elements Linear elasticity Basic concepts of plasticity and Mohr Coulomb model Non linear finite elements and solution techniques Applied theory: Introduction to PLAXIS T01 Tutorial: Soil testing tool, Mohr-Coulomb Applied theory: Shallow foundations Applied theory: Structural elements and interfaces T02 Tutorial: Shallow foundation Tentative schedule: Jan 16-20, 2012 Drained/undrained analysis Consolidation analysis Applied theory: Soil parameters for drained and undrained analysis Applied theory: Slope stability and phi-c reduction T03 Tutorial: Consolidation and phi-c reduction Critical state models Applied theory: Soil parameter for critical state models T04 Tutorial: Analysis of an embankment (inc stability) Tentative schedule May 28-May 21, 2012 Applied theory: Analysis of an embankment T05 Tutorial: Boston embankment (I) Hardening Soil Model and Small Strain Stiffness Applied theory: Soil parameters for Hardening Soil model Applied theory: Excavations T06 Tutorial: Excavation in Limburg Anisotropy, bonding and creep Applied theory: Numerical modelling of ground improvement T07 Tutorial: Boston embankment (II) Assessment: Coursework (100%) : Independent numerical analysis Part 1A: Identify a research paper with suitable numerical analysis Date of submission April 2, 2012 Part 1B: Numerical analysis and report Date of submission June 15, 2012 Recommended reading: Muir Wood, D Geotechnical Modelling Spon Press, 2004 Potts, D & Zdravkovic L Finite element analysis in geotechnical engineeringTheory Thomas Telford,1999 Potts, D & Zdravkovic L Finite element analysis in geotechnical engineeringApplication Thomas Telford,1999 Potts, D., Axelsson K., Grande, L Schweiger, H & Long, M Guidelines for the use of advanced numerical analysis Thomas Telford, 2002 Azizi, F Applied analysis in geotechnics E & F Spon, 2000 Muir Wood, D Soil behaviour and critical state soil mechanics Cambridge University Press,1990 Zienkiewicz & Taylor The Finite Element Method – available through various publishers PLUS selected research papers available Main aims of the module Give a comprehensive understanding of the role of soil modelling and numerical analysis in practical geotechnical context The focus is on: The selection of appropriate soil model considering a particular application and information available, Interpretation of values for soil parameters for numerical analysis, Idealisation and modelling of geotechnical problems with 2D finite element code PLAXIS Appreciation on the limitations of finite element modelling At the end of the course you will be competent (but not expert) on finite element modelling, and its opportunities and limitations, in geotechnical context Introduction to numerical modelling and finite elements Introduction Real problem σij, j + bi = Linear Momentum Balance North portal (Lleida) 360 εij = Terzaghi’s principle σ = σ′ + µρ 320 280 411+100 Results ( u j,i + ui ,j ) Strain displacement equation 412+000 Continuity equation Quaternary Colluvion Middle Eocene −∇T v = Early Eocene Limestone ∂εp Mathematical Model (PDE) Claystone & Siltstone σ′ = D ⋅ ε Darcy’s law v = −k ∇H 413+000 ∂t Mechanical constitutive lawMarl Anhydritic-Gypsiferous Claystone H= ρ T −r g γf m a.s.l 440 400 North portal (Lleida) 360 320 280 411+100 412+000 Quaternary Colluvion Solution Analytical Numerical Relevant 400 phenomena m a.s.l 440 Idealized problem Idealized problem(conceptual (conceptualmodel) model) Middle Eocene 413+000 Early Eocene Limestone Claystone & Siltstone Marl Anhydritic-Gypsiferous Claystone Introduction A rigorous solution must satisfy the following Equilibrium Compatibility Stress-strain relationship Boundary conditions Constitutive equations (cont.) In 3D the stress-strain equations take the form: σ yy ε x = (σ x − vσ y − vσ z ) E ε y = (σ y − vσ x − vσ z ) E ε z = (σ z − vσ x − vσ y ) E σ zz σ xx Constitutive equations (cont.) If a shear strain in applied to an elastic material, a shear strain is produced τ yx γ xy = τ xy G γ xy τ xy τ xy τ yx where G is the shear modulus Constitutive equations (cont.) Four elastic parameters are commonly used : • Young's modulus, E • Shear modulus, G, • Poisson's ratio ν • Bulk modulus, K An elastic material is fully specified, however, when values of two of these parameters are given E G= 2(1 + v) E K= 3(1 − 2v) 2D elastic analysis P z x To carry out FE analysis of 2D problems, it is necessary to specify the condition in the third dimension The plane strain condition is most commonly used in soil mechanics y P x y ε zz = 2D elastic analysis (cont.) In plane strain condition the out-of – plane strain is set to zero and Hooke’s law gives: x E ((1 − v)ε x + vε y ) (1 − 2v)(1 + v) E ((1 − v)ε y + vε x ) σy = (1 − 2v)(1 + v) τ xy = Gγ xy σx = y Drained and Undrained Analysis ∆u= ∆σ ∆σ’=0 Dissipation of ∆u with time Undrained Drained Consolidation ∆u=0 ∆σ’= ∆σ Drained and Undrained Analysis (cont.) (a) The shear modulus is identical for drained and undrained loading Gu = G ′ = G (b) The drained and undrained Young's moduli are related by the expression: E′ = (1 + v ′) E u Note that for most soils the value ν’ of generally lies in the range 0.3 to 0.35 for sands and 0.2-0.3 for clays Young’s modulus values, however, may vary substantially between different materials and stress levels Isotropic elasticity in 3D σ ′y τ yz τ yx τ xy σ x′ τ zy y τ xz τ zx z x σ z′ ε x = (σ ' x −v'σ ' y −v'σ ' z ) E' ε y = (σ ' y −v'σ ' x −v'σ ' z ) E' ε z = (σ ' z −v'σ ' x −v'σ ' y ) E' γ xy = γ yz = γ zx = τ xy G' τ yz G' τ zx G' Cross-isotropic elasticity (around y-axis) Sampling direction h v τ yz σ ′y τ yx εx = τ xy σ x′ τ zy y τ xz τ zx z x σ z′ General 3D elasticity would require the specification of 21 elastic constants! σ 'x Eh ' − vhh ' v ' σ ' y − vh σ ' z Ev ' Ev ' σ ' y vhh ' vhh ' εy = − σ 'x + − σ 'z Ev ' E v ' Ev ' vvh ' vhh ' σ 'z εz = σ 'x − σ 'y + Eh ' Ev ' Eh ' γ xy = γ yz = γ zx = τ xy Gvh ' τ yz Gvh ' τ zx Ghh ' Needs elastic constants! Oedometer test Load y Soil Lateral strains are zero, therefore measure modulus is not Young’s modulus Based on Hooke’s law: Eoed ′ = (1 − v ′) E ′ (1 − 2v ′)(1 + v ′) Vertical strain εyy mv = / E ' oed E'oed Unload/reload path E'oed,ur Primary compression σ ′yy Triaxial test (see also Muir Wood 1990) Deviator stress q E'ur qf E'50 1 qf / Axial strain, εa Shearing with constant cell pressure Secant or Tangent E? σ'1 σ'1 Et Young’s modulus E’ Es ε1 ε1 Poisson’s ratioν’ − < ν ' < G' = E' 2(1 +ν ' ) K'= E' 3(1 − 2ν ' ) Non-Linear Elasticity Non linear elasticity e ln(p') Bulk modulus K’ K'= κ dp ' dp ' (1 + e) p ' = (1 + e) = dε v de κ Shear modulus G’ G' = 3(1 − 2ν ' ) K' 2(1 +ν ' ) with − < ν ' < Elasticity vs Plasticity In elasticity, there is a one-to-one relationship between stress and strain Such a relationship may be linear or non-linear An essential feature is that the application and removal of a stress leaves the material in its original condition Elasticity vs Plasticity for elastic materials, the mechanism of deformation depends on the stress increment for plastic materials which are yielding, the mechanism of (plastic) deformation depends on the stress reversible = elastic irreversible = plastic Next lecture will look at the Basic Concepts of Plasticity and Mohr Coulomb model