PLAXIS Version 8 Material Models ManualTABLE OF CONTENTS 1 Introduction ..................................................................................................1-1 1.1 On the use of different models ...............................................................1-1 1.2 Limitations .............................................................................................1-3 Preliminaries on material modelling ..........................................................2-1 2.1 General definitions of stress...................................................................2-1 2.2 General definitions of strain...................................................................2-4 2.3 Elastic strains .........................................................................................2-5 2.4 Undrained analysis with effective parameters .......................................2-8 2.5 Undrained analysis with undrained parameters ...................................2-13 2.6 The initial preconsolidation stress in advanced models .......................2-13 2.7 On the initial stresses ...........................................................................2-14 The Mohr-Coulomb model (perfect-plasticity) ........................................3-1 3.1 Elastic perfectly-plastic behaviour.........................................................3-1 3.2 Formulation of the Mohr-Coulomb model.............................................3-3 3.3 Basic parameters of the Mohr-Coulomb model .....................................3-5 3.4 Advanced parameters of the Mohr-Coulomb model..............................3-8 The Jointed Rock model (anisotropy) .......................................................4-1 4.1 Anisotropic elastic material stiffness matrix..........................................4-2 4.2 Plastic behaviour in three directions ......................................................4-4 4.3 Parameters of the Jointed Rock model...................................................4-7 The Hardening-Soil model (isotropic hardening) ....................................5-1 5.1 Hyperbolic relationship for standard drained triaxial test ......................5-2 5.2 Approximation of hyperbola by the Hardening-Soil model...................5-3 5.3 Plastic volumetric strain for triaxial states of stress ...............................5-6 5.4 Parameters of the Hardening-Soil model ...............................................5-7 5.5 On the cap yield surface in the Hardening-Soil model ........................5-11 Soft-Soil-Creep model (time dependent behaviour) .................................6-1 6.1 Introduction............................................................................................6-1 6.2 Basics of one-dimensional creep............................................................6-2 6.3 On the variables τc and εc ......................................................................6-4 6.4 Differential law for 1D-creep.................................................................6-6 6.5 Three-dimensional-model ......................................................................6-8 6.6 Formulation of elastic 3D-strains.........................................................6-11 6.7 Review of model parameters................................................................6-12 2 3 4 5 6 .............................2 Application of the Hardening-Soil model on real soil tests ..........2 Yield function for triaxial stress state (σ2' = σ3') ................................ 8-24 User-defined soil models ...........1 Isotropic states of stress and strain (σ1' = σ2' = σ3') ......... 8-1 8..... 9-10 References..8 Validation of the 3D-model ... 8-1 8.....................................................................3 Input of UD model parameters via user-interface........................................ 7-1 7........................................... 9-1 9...............................3 SSC model: response in one-dimensional compression test .................................. 10-1 8 9 10 Appendix A ...............5 SS model: response in isotropic compression test ...6 Submerged construction of an excavation with HS model ............Creating a debug-file for User-Defined soil models ............................................ 7-5 Applications of advanced soil models .....7 Road embankment construction with the SSC model....... 7-1 7................................................6................................1 Introduction ..................... 8-22 8..4 SSC model: undrained triaxial tests at different loading rates ..................... 8-12 8.............................................Symbols Appendix B ............... 8-17 8.....2 Implementation of UD models in calculations program ............... 8-20 8..................... 8-5 8... 9-1 9..... 7-3 7..................... 9-1 9................................1 HS model: response in drained and undrained triaxial tests ..................... 6-16 7 The Soft Soil model ..................................Fortran subroutines for User-Defined soil models Appendix C .....................................3 Parameters of the Soft-Soil model ............................... Besides the five model parameters mentioned above. As it involves only two input parameters. but both very soft and very stiff soils tend to give other ratios of Eoed / E50. initial soil conditions play an essential role in most soil deformation problems. however. the triaxial unloading stiffness. In contrast to the Mohr-Coulomb model. Initial horizontal soil stresses have to be generated by selecting proper K0-values. ϕ and c for soil plasticity and ψ as an angle of dilatancy. Reduced elastic properties may be defined for the stratification direction. Due to this constant stiffness. However. it is generally too crude to capture essential features of soil and rock behaviour. Hardening-Soil model (HS) The Hardening-Soil model is an advanced model for the simulation of soil behaviour. 1. This MohrCoulomb model represents a 'first-order' approximation of soil or rock behaviour. It is recommended to use this model for a first analysis of the problem considered. may be thought of as the simplest available stress-strain relationship. Eoed. As for the Mohr-Coulomb model. especially meant to simulate the behaviour of rock layers involving a stratification and particular fault directions. limiting states of stress are described by means of the friction angle. This means that all stiffnesses increase with . Hooke's law of linear. the Hardening-Soil model also accounts for stress-dependency of stiffness moduli. ν. Jointed Rock model (JR) The Jointed Rock model is an anisotropic elastic-plastic model. E and ν for soil elasticity. isotropic elasticity.e.1 ON THE USE OF DIFFERENT MODELS Mohr-Coulomb model (MC) The elastic-plastic Mohr-Coulomb model involves five input parameters. computations tend to be relatively fast and one obtains a first impression of deformations. For modelling massive structural elements and bedrock layers. i. For each layer one estimates a constant average stiffness. E50. The intact rock is considered to behave fully elastic with constant stiffness properties E and ν. i. we have Eur ≈ 3 E50 and Eoed ≈ E50. Young's modulus. and the dilatancy angle. Each plane has its own strength parameters ϕ and c. ϕ. linear elasticity tends to be appropriate. and Poisson's ratio. E. the cohesion.1 INTRODUCTION The mechanical behaviour of soils may be modelled at various degrees of accuracy. c. ψ. Plasticity can only occur in a maximum of three shear directions (shear planes).e. and the oedometer loading stiffness. soil stiffness is described much more accurately by using three different input stiffnesses: the triaxial loading stiffness. Eur. As average values for various soil types. for example. i. all three input stiffnesses relate to a reference stress. In many cases.e. No doubt. one seldom has test results from both triaxial and oedometer tests. but it tends to pay off. Although the modelling capabilities of this model are superceded by the Hardening Soil model. all soils exhibit some creep and primary compression is thus followed by a certain amount of secondary compression. creep and stress relaxation. . Soft Soil model (SS) The Soft Soil model is a Cam-Clay type model especially meant for primary compression of near normally-consolidated clay-type soils. but it does not account for viscous effects. as normally encountered in tunnelling and other excavation problems. because existing PLAXIS users might be comfortable with this model and still like to use it in their applications. embankments. In fact. For unloading problems.e. proper initial soil conditions are also essential when using the Soft-Soil-Creep model. the Soft Soil model is still retained in the current version. i. etc. Soft-Soil-Creep model (SSC) The above Hardening-Soil model is suitable for all soils. Hence. silts and peat. The above idea of analyzing geotechnical problems with different soil models may seem costly. Analyses with different models It is advised to use the Mohr-Coulomb model for a relatively quick and simple first analysis of the problem considered. as these models account for the effect of overconsolidation.e. Finally. and it is appropriate to use the Hardening-Soil model in an additional analysis. The latter is most dominant in soft soils. and we thus implemented a model under the name Soft-Soil-Creep model. Please note that the Soft-Soil-Creep model is a relatively new model that has been developed for application to settlement problems of foundations. First of all due to the fact that the Mohr-Coulomb analysis is relatively quick and simple and secondly as the procedure tends to reduce errors. a Soft-Soil-Creep analysis can be performed to estimate creep. the Soft-Soil-Creep model hardly supersedes the simple Mohr-Coulomb model. i. As for the Mohr-Coulomb model. being usually taken as 100 kPa (1 bar). but good quality data from one type of test can be supplemented by data from correlations and/or in situ testing.pressure. For the Hardening-Soil model and the Soft-Soil-Creep model this also includes data on the preconsolidation stress. there is no use in further more advanced analyses. When good soil data is lacking. secondary compression in very soft soils. normally consolidated clays. one has good data on dominant soil layers. the use of the Hardening Soil model generally results in longer calculation times. In such cases . In order to model cyclic loading with good accuracy one would need a more complex model. ψ. but is kept for users who are familiar with this model. and the ability to judge the reliability of the computational results. which implicitly involves some inevitable numerical and modelling errors. This is especially the case for excavation problems.5.2 LIMITATIONS The PLAXIS code and its soil models have been developed to perform calculations of realistic geotechnical problems. The utilization of the SS-model should be limited to the situations that are dominated primarily by compression. Although much care has been taken for the development of the PLAXIS code and its soil models. Interfaces Interface elements are generally modelled by means of the bilinear Mohr-Coulomb model. It is certainly not recommended for use in excavation problems. the simulation of reality remains an approximation. the interface element will only pick up the relevant data (c. including tunnelling. as described in Section 3. SSC-model All above limitations also hold true for the Soft-Soil-Creep model. As a final remark.1. Both the soil models and the PLAXIS code are constantly being improved such that each new version is an update of the previous ones.2 of the Reference Manual. ν) for the MohrCoulomb model. Moreover. In this respect PLAXIS can be considered as a geotechnical simulation tool. In fact the SS-model is superceded by the HS-model. Some of the present limitations are listed below: HS-model It is a hardening model that does not account for softening due to soil dilatancy and debonding effects. The soil models can be regarded as a qualitative representation of soil behaviour whereas the model parameters are used to quantify the soil behaviour. the accuracy at which reality is approximated depends highly on the expertise of the user regarding the modelling of the problem. SS-model The same limitations (including these in the SSC-model) hold in the SS-model. since the material stiffness matrix is formed and decomposed in each calculation step. ϕ. In fact. the selection of model parameters. E. When a more advanced model is used for the corresponding cluster material data set. the understanding of the soil models and their limitations. In addition this model tends to over predict the range of elastic soil behaviour. it is an isotropic hardening model so that it models neither hysteretic and cyclic loading nor cyclic mobility. following a power law with Eur proportional to σm. Hence. E = Eur where Eur is stress level dependent. . the power m is equal to 1 and Eur is largely determined by the swelling constant κ*. For the SoftSoil-Creep model.the interface stiffness is taken to be the elastic soil stiffness. Later sections focus on initial conditions for advanced material models. σyz = σzy and σzx = σxz. in Version 8 only plane strain and axisymmetric conditions are considered. σ yz = σ zx = 0 According to Terzaghi's principle. > Element and material model formulations in PLAXIS are fully three dimensional. In the ! ′ . In subsequent sections the basic stress-strain relationship is formulated and the influence of pore pressures in undrained materials is described. As a result. σ following section it is described how stresses and strains are defined in PLAXIS. Material models are often expressed in a form in which infinitesimal increments of stress (or 'stress rates') are related to infinitesimal increments of strain (or 'strain rates'). and the strain rates.1) In the standard deformation theory. All material models implemented in the PLAXIS are based on a ! . stresses in the soil are divided into effective stresses.2) In plane strain condition.1 GENERAL DEFINITIONS OF STRESS Stress is a tensor which can be represented by a matrix in Cartesian coordinates: σ xx σ = σ yx σ zx σ xy σ yy σ zy σ xz σ yz σ zz (2. effective shear stresses are equal to total shear stresses. σ'. ε relationship between the effective stress rates. which involve only six different components: σ = (σ xx σ yy σ zz σ xy σ yz σ zx )T (2. however. σ w: σ = σ '+σ w (2. and pore pressures. Positive normal stress components are considered to . stresses are often written in vector notation.2 PRELIMINARIES ON MATERIAL MODELLING A material model is a set of mathematical equations that describes the relationship between stress and strain. In this situation. However.3) Water is considered not to sustain any shear stresses. the stress tensor is symmetric such that σxy = σyx . 2. in fact. models are often presented with reference to the principal stress space.6) Hence. Material models for soil and rock are generally expressed as a relationship between infinitesimal increments of effective stress and infinitesimal increments of strain.1 General three-dimensional coordinate system and sign convention for stresses It is often useful to use principal stresses rather than Cartesian stress components when formulating material models. In this manual. In such a relationship.e. as indicated in Figure 2.4) σyy σyz σzy σzz σzx σ xz σyx σxy σxx x z Figure 2. . σ'2. This equation gives three solutions for σ'. Principal stresses are the stresses in such a coordinate system direction that all shear stress components are zero. whereas negative normal stress components indicate pressure (or compression). σ'3). In PLAXIS the principal effective stresses are arranged in algebraic order: σ'1 ≤ σ'2 ≤ σ'3 (2. the eigenvalues of the stress tensor. the principal effective stresses (σ'1. i.represent tension. Principal effective stresses can be determined in the following way: det σ '−σ ' I = 0 ( ) (2.5) where I is the identity matrix. Principal stresses are.2. infinitesimal increments of effective stress are represented by stress rates (with a dot above the stress symbol): ! ' = (σ ! ' xx σ ! ' yy σ ! ' zz σ ! xy σ y ! yz σ ! zx )T σ (2. σ'1 is the largest compressive principal stress and σ'3 is the smallest compressive principal stress. which is defined as: 27 J 3 θ=1 arcsin 3 3 q3 (2.8c) ′ = p′ + 2 θ) -σ 2 3 q sin ( 2 ′ = p′ + 2 -σ3 3 q sin θ + 3 π ( in which θ is referred to as Lode's angle (a third invariant).7b) − σ' xx )2 + 6 σ xy + σ yz + σ zx ( )) where p' is the isotropic effective stress. has the important property that it reduces to q = │σ1'-σ3'│ for triaxial stress states with σ2' = σ3'. which are stress measures that are independent of the orientation of the coordinate system.7a) ((σ' xx −σ ′ yy ) + (σ' 2 yy − σ' zz ) + (σ' 2 zz 2 2 2 (2.-σ'1 -σ'1 = -σ'2 = -σ'3 -σ'3 -σ ' 2 Figure 2.1 3 3 q= 1 2 ( ) (2.2 Principal stress space In addition to principal stresses it is also useful to define invariants of stress. Two useful stress invariants are: ′ +σ3 ′) (σ 1′ + σ 2 σ ' xx +σ ' yy +σ ' zz = − 1 p′ = . or mean effective stress. Note that the convention adopted for p' is positive for compression in contrast to other stress measures. q. The equivalent shear stress. and q is the equivalent shear stress.8a) (2.9) .8b) (2. Principal effective stresses can be written in terms of the invariants: 2 ′ = p′ + 2 .σ1 3 q sin θ − 3 π ( ) ) (2. with 2 2 ′ ′ ′ σ′ ′ σ xx ′ -p′)σ yz ′ 2 σ′ ′ σ xy + σ′ + 2σ xy σ yzσ zx J3 = (σ ′ xx-p ) σ yy-p ( zz -p ) + ( yy-p σ zx + ( zz -p ) ( ) ( ) (2. only the sum of complementing Cartesian shear strain components εij and εji result in shear stress.13d) γ yz = ε yz + ε zy = (2. Under the above conditions.12) (2. εzy. instead of εxy. This sum is denoted as the shear strain γ. strains are often written in vector notation.13b) ε zz = (2.2 GENERAL DEFINITIONS OF STRAIN Strain is a tensor which can be represented by a matrix with Cartesian coordinates as: ε xx ε = ε yx ε zx ε xy ε yy ε zy ε xz ε yz ε zz (2. Hence.11) According to the small deformation theory. εyx.13e) . which involve only six different components: ε = (ε xx ε yy ε xx = ∂u x ∂x ε zz γ xy γ yz γ zx )T (2.13a) ε yy = ∂u y ∂y ∂u z ∂z (2. εyz .13c) γ xy = ε xy + ε yx = ∂u x ∂u y + ∂y ∂x ∂u y ∂z + ∂u z ∂y (2. εzx and εxz the shear strain components γxy.10) 2. γyz and γzx are used respectively. For elastoplastic models. as considered in PLAXIS version 8. 1 ε zz = u x and γ xz = γ yz = 0 (r = radius) r In analogy to the invariants of stress. positive normal strain components refer to extension. where infinitesimal increments of strain are considered.3 ELASTIC STRAINS Material models for soil and rock are generally expressed as a relationship between infinitesimal increments of effective stress ('effective stress rates') and infinitesimal increments of strain ('strain rates').16) Throughout this manual.γ zx = ε zx + ε xz = ∂u z ∂u x + ∂x ∂z (2. which is defined as the sum of all normal strain components: ε v = ε xx + ε yy + ε zz = ε 1 + ε 2 + ε 3 (2. the superscript e will be used to denote elastic strains and the superscript p will be used to denote plastic strains.14) for plane strain conditions. ε zz = γ xz = γ yz = 0 whereas for axisymmetric conditions.15) The volumetric strain is defined as negative for compaction and as positive for dilatancy.13f) Similarly as for stresses. εν. ! = (ε !xx ε ! yy ε !zz ε !xy γ ! yz γ !zx γ ) T (2. whereas negative normal strain components indicate compression. A strain invariant that is often used is the volumetric strain. as used in PLAXIS program. In the formulation of material models. This relationship may be expressed in the form: . these increments are represented by strain rates (with a dot above the strain symbol). strains are decomposed into elastic and plastic components: ε = εe + ε p (2. 2. it is also useful to define invariants of strain. unless a different meaning is explicitly stated. Entering a particular value for one of the alternatives G or Eoed results in a change of the E modulus. Two parameters are used in this model. Note that in this type of approach. pore-pressures are explicitly excluded from the stress-strain relationship. A stiffness modulus may also be indicated with the subscript ref to emphasize that it refers to a particular reference level (yref) (see further). such as the shear modulus G. This model is available under the name Linear Elastic model.! !′ = M ε σ (2. but it is also the basis of other models. the effective Young's modulus. ν'. In the remaining part of this manual effective parameters are denoted without dash ('). The symbols E and ν are sometimes used in this manual in combination with the subscript ur to emphasize that the parameter is explicitly meant for unloading and reloading.19b) E oed = (1 -ν )E ( 1 − 2ν )(1 + ν ) (2. the bulk modulus K. is given by: G= K= E 2(1 + ν ) E 3(1 − 2ν ) (2.17) M is a material stiffness matrix. . and the effective Poisson's ratio.19). The simplest material model in PLAXIS is based on Hooke's law for isotropic linear elastic behaviour.18) The elastic material stiffness matrix is often denoted as De. and the oedometer modulus Eoed. Hooke's law can be given by the equation: ! 'xx ν' 1−ν ' ν ' σ σ ! ν ' 1−ν ' ν ' 'yy ν' σ ! 'zz ν ' 1 −ν ' E' = ! 'xy (1− 2ν ')(1 +ν ') 0 0 0 σ σ !' 0 0 0 yz 0 0 ! 'zx σ 0 0 0 0 1 −ν ' 2 0 0 0 0 0 0 1 −ν ' 2 0 !xx 0 ε ! 0 ε yy !zz 0 ε !xy 0 γ !yz 0 γ 1 −ν ' !zx γ 2 (2. calculated from Eq. The relationship between Young's modulus E and other stiffness moduli.19c) During the input of material parameters for the Linear Elastic model or the MohrCoulomb model the values of G and Eoed are presented as auxiliary parameters (alternatives). E'.19a) (2. (2. Note that the alternatives are influenced by the input values of E and ν. Figure 2.3 Parameters tab for the Linear Elastic model It is possible for the Linear Elastic model to specify a stiffness that varies linearly with depth.3.4. Here one may enter a value for Eincrement which is the increment of stiffness per unit of depth. as shown in Figure 2. This can be done by entering the advanced parameters window using the Advanced button.4 Advanced parameter window . as indicated in Figure 2. Figure 2. but it is of interest to simulate structural behaviour. it is explained how PLAXIS deals with this special option.20) The Linear Elastic model is usually inappropriate to model the highly non-linear behaviour of soil. water is supposed not to sustain any shear stress. Above yref the stiffness is equal to Eref.21f) σ yy = σ ' yy +σ w σ zz = σ ' zz +σ w σ xy = σ ' xy σ yz = σ ' yz σ zx = σ ' zx Note that. . The presence of pore pressures in a soil body. For these applications.21a) (2. 2. In this Section. for which strength properties are usually very high compared with those of soil.21e) (2. Below the stiffness is given by: Eactual = Eref + yref − y Eincrement ( ) y < yref (2. σw is considered negative for pressure. such as thick concrete walls or plates.21b) (2. contributes to the total stress level. According to Terzaghi's principle. 2. total stresses σ can be divided into effective stresses σ' and pore pressures σw (see also Eq. However.Together with the input of Eincrement the input of yref becomes relevant.4 UNDRAINED ANALYSIS WITH EFFECTIVE PARAMETERS In the PLAXIS it is possible to specify undrained behaviour in an effective stress analysis using effective model parameters. usually caused by water. the Linear Elastic model will often be selected together with Non-porous type of material behaviour in order to exclude pore pressures from these structural elements. and therefore the effective shear stresses are equal to the total shear stresses: σ xx = σ ' xx +σ w (2. similar to the total and the effective stress components.21d) (2. This is achieved by identifying the Type of material behaviour (Material type) of a soil layer as Undrained.3).21c) (2. pexcess: σw = psteady + pexcess (2. it follows: ! excess !w = p σ Hooke's law can be inverted to obtain: e ! 'xx ε !xx 0 0 0 σ 1 −ν ' −ν ' e ! ' yy !yy 0 0 0 σ ε −ν ' 1 −ν ' e ε ! 'zz !zz 0 0 0 σ 1 −ν ' −ν ' 1 e = ! xy 0 0 2 + 2ν ' 0 0 σ !xy E' 0 γ e γ ! yz σ 0 0 0 0 2 + 2ν ' 0 ! yz e ! zx !zx 0 0 0 0 2 + 2ν ' 0 σ γ (2. Excess pore pressures are generated during plastic calculations for the case of undrained material behaviour.e. (2.26) .24) Substituting Eq.22) Steady state pore pressures are considered to be input data. This generation of steady state pore pressures is discussed in Section 3. Undrained material behaviour and the corresponding calculation of excess pore pressures is described below. (2. psteady.25) Considering slightly compressible water. generated on the basis of phreatic levels or groundwater flow. the rate of pore pressure is written as: ! w = K w (ε σ !e !e !e xx + ε yy + ε zz ) n in which Kw is the bulk modulus of the water and n is the soil porosity.21) gives: e ! xx −σ !w ε !xx 0 0 0 σ 1 −ν ' −ν ' e ! yy −σ !w !yy 0 0 0 σ ε −ν ' 1 −ν ' e ε ! zz −σ !w !zz 0 0 0 σ 1 −ν ' −ν ' 1 e = ! xy 0 0 2 + 2ν ' 0 0 σ !xy E' 0 γ γ 0 ! yz 0 0 0 2 + 2ν ' 0 σ !e yz e ! zx !zx 0 0 0 0 2 + 2ν ' 0 σ γ (2.8 of the Reference Manual. i. Since the time derivative of the steady state component equals zero.A further distinction is made between steady state pore stress. and excess pore stress.23) (2. 29) Hence. the special option for undrained behaviour in PLAXIS is such that the effective parameters G and ν are transferred into undrained parameters Eu and νu according to Eq.495 − ν ′ K w = 3( K ′ = 300 K ′ > 30 K ′ (1 − 2ν u )(1 + ν ′) 1 +ν ′ n (2. Kw >>n K'. In order to avoid numerical problems caused by an extremely low compressibility. In order to ensure realistic computational results. Hence.5 leads to singularity of the stiffness matrix.e. but a realistic bulk modulus for water is very large.35. Note that the index u is used to indicate auxiliary parameters for undrained soil. In fact.22). i. water is not fully incompressible. taking νu = 0.5.495. However. for undrained material behaviour a bulk modulus for water is automatically added to the stiffness matrix.28) µ= 1 Kw 3 n K' (2. Consequently. The value of the bulk modulus is given by: ν u − ν ′) 0.21) and (2. νu is by default taken as 0. Users will get a warning as soon as larger Poisson's ratios are used in combination with undrained material behaviour. This condition is sufficiently ensured by requiring ν' ≤ 0. (2. Fully incompressible behaviour is obtained for νu = 0. the bulk modulus of the water must be high compared with the bulk modulus of the soil skeleton.The inverted form of Hooke's law may be written in terms of the total stress rates and the undrained parameters Eu and νu: e ! 'xx ε 0 0 0 σ !xx 1 −ν u −ν u e ! ' yy 0 0 0 σ !yy ε −ν u 1 −ν u e ε ! 'zz 0 0 0 σ !zz 1 −ν u −ν u 1 e = ! xy 0 0 2 + 2ν u 0 0 σ !xy Eu 0 γ e γ ! yz σ 0 0 0 0 2 + 2ν u 0 ! yz e ! zx 0 0 0 0 2 + 2ν u !zx 0 σ γ (2. which makes the undrained soil body slightly compressible. Eu and νu should not be confused with Eur and νur as used to denote unloading / reloading.27) where: Eu = 2 G ( 1 + ν u ) νu = K′ = ν '+ µ ( 1 + ν ') 1 + 2 µ (1 + ν ') E' 3(1 − 2ν ′) (2.30) . ϕ' and not Eu. c'.at least for ν' ≤ 0. and the porosity. νu. effective stresses and excess pore pressures (see Undrained behaviour): Total stress: Effective stress: Excess pore pressure: ∆ p = K u ∆ εν ∆p′ = (1 − B )∆p = K ′∆εν ∆pw = B∆p = Kw ∆εν n Note that effective model parameters should be entered in the material data set. ν'. Kw. ref n = Ku − K ' where K'= E' 3(1 − 2ν ' ) This value of Kw. The undrained bulk modulus is automatically calculated by PLAXIS using Hooke's law of elasticity: Ku = and or 2G (1 + ν u ) 3(1 − 2ν u ) where G= E' 2(1 + ν ' ) ν u = 0. Kw0 (= 2⋅106 kN/m2). ϕu .35.ref / n: K w.495 νu = 3ν '+ B (1 − 2ν ' ) 3 − B (1 − 2ν ' ) (when using the Standard setting) (when using the Manual setting) A particular value of the undrained Poisson's ratio. cu (su).e. but the degree of saturation. If the value of Skempton's B-parameter is unknown. Skempton B-parameter: When the Material type (type of material behaviour) is set to Undrained. Ku.ref / n is generally much smaller than the real bulk stiffness of pure water. E'. νu. i. the bulk stiffness of the pore fluid can be estimated from: . S. are known instead. PLAXIS automatically assumes an implicit undrained bulk modulus. for the soil as a whole (soil skeleton + water) and distinguishes between total stresses. n. In retrospect it is worth mentioning here a review about the Skempton B-parameter. implies a corresponding reference bulk stiffness of the pore fluid. 31) The type of elements used in the PLAXIS are sufficiently adequate to avoid mesh locking effects for nearly incompressible materials. For soft soil projects. in situ tests and laboratory tests may have been performed to obtain undrained soil parameters.0 Kw Kw K air 1 = 0 n SK air + ( 1 − S)K w n where K' = E' 3( 1 − 2ν') and Kair = 200 kN/m2 for air under atmospheric pressure. This special option to model undrained material behaviour on the basis of effective model parameters is available for all material models in the PLAXIS program. In such situations measured undrained Young's moduli can be easily converted into effective Young's moduli by: E′ = 2(1 + ν ') Eu 3 (2. cannot easily be used to determine the effective strength parameters ϕ and c. with explicit distinction between effective stresses and (excess) pore pressures. Such an analysis requires effective soil parameters and is therefore highly convenient when such parameters are available. This option is only available for the Mohr-Coulomb model and the Hardening-Soil model. effective values must be entered for the elastic parameters E and ν ! . Note that whenever the Material type parameter is set to Undrained. Instead. For such projects PLAXIS offers the possibility of an undrained analysis with direct input of the undrained shear strength (cu or su) and ϕ = ϕu = 0°. according to: !w = σ Kw !v ε n (2. This enables undrained calculations to be executed with effective input parameters. however. The value of Skempton's Bparameter can now be calculated from the ratio of the bulk stiffnesses of the soil skeleton and the pore fluid: B= 1 nK ' 1+ Kw The rate of excess pore pressure is calculated from the (small) volumetric strain rate. but not for the Soft Soil (Creep) model. accurate data on effective parameters may not always be available.32) Undrained shear strengths. 34) These two ways of specifying the vertical preconsolidation stress are illustrated in Figure 2. the ratio of the greatest vertical stress previously reached. Note that this type of approach is not possible when using the Soft Soil Creep model. The Pre-Overburden Pressure is defined by: POP = | σ p .5 UNDRAINED ANALYSIS WITH UNDRAINED PARAMETERS If. Hence. an effective stress analysis using the Undrained option in PLAXIS to simulate undrained behaviour is preferable over a total stress analysis. σyy'0 . . In general. ppeq to determine the initial position of a cap-type yield surface.5. for any reason.495 in combination with the undrained strength properties c = cu and ϕ = ϕu = 0°. In graphical output of stresses the stresses in Non-porous clusters are not plotted. σp. If a material is overconsolidated. information is required about the Over-Consolidation Ratio (OCR). In the engineering practice it is common to use a vertical preconsolidation stress. OCR = σp σ '0 yy (2. and the in-situ effective vertical stress. one may simulate undrained behaviour by selecting the Non-porous option and directly entering undrained elastic properties E = Eu and ν = νu = 0.2. 2. but PLAXIS needs an equivalent isotropic preconsolidation stress. all tabulated output referring to effective stresses should now be interpreted as total stresses and all pore pressures are equal to zero.σ '0 yy | (2.e.5).6 THE INITIAL MODELS PRECONSOLIDATION STRESS IN ADVANCED When using advanced models in PLAXIS an initial preconsolidation stress has to be determined. If one does want graphical output of stresses one should select Drained instead of Non-porous for the type of material behaviour and make sure that no pore pressures are generated in these clusters. i. In this case a total stress analysis is performed without distinction between effective stresses and pore pressures.33) It is also possible to specify the initial stress state using the Pre-Overburden Pressure (POP) as an alternative to prescribing the overconsolidation ratio. σp (see Figure 2. it is desired not to use the Undrained option in PLAXIS to perform an undrained analysis. 2.35) Where K 0 is the K 0 -value associated with normally consolidated states of stress. but differences with the Jaky correlation are modest. preloaded to σyy'=σp and subsequently unloaded to σyy' = σyy'0.5a. given by (see Figure 2. Consider a one-dimensional compression test. During unloading the sample behaves elastically and the incremental stress ratio is. The procedure that is followed here is described below. The calculation of ppeq is based on the stress state: σ'1 = σ p NC and: NC σ' 2 = σ'3 = K 0 σp (2.7 ON THE INITIAL STRESSES In overconsolidated soils the coefficient of lateral earth pressure is larger than for normally consolidated soils. Using OCR.OCR = σp σ'0 yy POP 0 σ 'yy σp 0 σ'yy σp Figure 2. This effect is automatically taken into account for advanced soil models when generating the initial stresses using the K0-procedure. For the Hardening-Soil model the default parameter settings is such that we have the Jaky NC formula K 0 ≈ 1 − sin φ .6): 0 0 0 NC NC ∆σ ' xx K 0 σ p − σ' xx K 0 OCRσ' yy − σ' xx ν ur = = = 0 0 ∆σ ' yy 1 − ν ur (OCR − 1)σ' yy σ p − σ' yy (2. For the Soft-Soil-Creep model.36) . according to Hooke's law. the default setting is slightly different. Using POP The pre-consolidation stress σp is used to compute ppeq which determines the initial position of a cap-type yield surface in the advanced soil models.5 Illustration of vertical preconsolidation stress in relation to the in-situ vertical stress. 2. 2.5b. as generally observed in overconsolidated soils. the stress ratio of the overconsolidated soil sample is given by: σ '0 νur xx NC (OCR .37) is only valid in the elastic domain. (2.37) The use of a small Poisson's ratio.where K0nc is the stress ratio in the normally consolidated state.1) = K0 OCR − 0 1 − νur σ ' yy (2. Note that Eq.6 Overconsolidated stress state obtained from primary loading and subsequent unloading . because the formula was derived from Hooke's law of elasticity. the stress ratio will be limited by the MohrCoulomb failure condition. as discussed previously. If a soil sample is unloaded by a large amount. Hence. resulting in a high degree of overconsolidation. -σ'yy -σp K0NC 1 -σ'yy0 1-νur νur -σ'xx0 -σ'xx Figure 2. will lead to a relatively large ratio of lateral stress and vertical stress. . A perfectly-plastic model is a constitutive model with a fixed yield surface. a plastic potential function g is introduced. f. The case g ≠ f is denoted as non-associated plasticity. the plastic strain rates are written as: !p = λ ε ∂g ∂ σ’ (3.3) in which λ is the plastic multiplier.2) According to the classical theory of plasticity (Hill. a yield surface that is fully defined by model parameters and not affected by (plastic) straining. In order to evaluate whether or not plasticity occurs in a calculation.e. a yield function.3 THE MOHR-COULOMB MODEL (PERFECT-PLASTICITY) Plasticity is associated with the development of irreversible strains. for Mohr-Coulomb type yield functions. This classical form of the theory is referred to as associated plasticity. i. in addition to the yield function.1) into Hooke's law (2.4b) .18) leads to: e e e ! −ε !’ = D ε ! = D ε !p σ ( ) (3. plastic strain rates are proportional to the derivative of the yield function with respect to the stresses. For stress states represented by points within the yield surface. (3. However. For purely elastic behaviour λ is zero.4a) for: f=0 and: (Plasticity) (3. A yield function can often be presented as a surface in principal stress space. 3. Therefore.1) Hooke's law is used to relate the stress rates to the elastic strain rates. is introduced as a function of stress and strain. In general. Substitution of Eq. the theory of associated plasticity leads to an overprediction of dilatancy. the behaviour is purely elastic and all strains are reversible. 1950). whereas in the case of plastic behaviour λ is positive: λ =0 λ >0 for: f<0 or: ∂f T e D ε ! ≤ 0 ∂ σ’ ∂ f e !>0 D ε ∂ σ’ T (Elasticity) (3.1 ELASTIC PERFECTLY-PLASTIC BEHAVIOUR The basic principle of elastoplasticity is that strains and strain rates are decomposed into an elastic part and a plastic part: ε = εe + ε p ! =ε !e + ε !p ε (3. This means that the plastic strain rates can be represented as vectors perpendicular to the yield surface. as defined by Eq. 1984): e α e ∂g ∂ f T e ! !’ = D − D D ε σ ∂ σ’ ∂ σ’ d where: (3. Vermeer & de Borst.).4b).. ∂ σ’ ∂ σ’ (3. If the material behaviour is elastic. whilst for plasticity. λ2.. .. several quasi independent yield functions (f1. the value of α is equal to zero.5b) The parameter α is used as a switch. as defined by Eq. . (3.4a)..1 Basic idea of an elastic perfectly plastic model These equations may be used to obtain the following relationship between the effective stress rates and strain rates for elastoplasticity (Smith & Griffith. For such a yield surface the theory of plasticity has been extended by Koiter (1960) and others to account for flow vertices involving two or more plastic potential functions: ! p = λ1 ε ∂ g1 ∂ g2 + λ2 + . the value of α is equal to unity.. The above theory of plasticity is restricted to smooth yield surfaces and does not cover a multi surface yield contour as present in the Mohr-Coulomb model.6) Similarly. . 1982. f2.σ’ ε Figure 3.) are used to determine the magnitude of the multipliers (λ1..5a) d= ∂ f e ∂ g D ∂ σ’ ∂ σ’ T (3. (3. 1982): f1a = f1b = f 2a = f 2b = f 3a = f 3b = 1 2 1 2 1 2 1 2 1 2 1 2 (σ '2 −σ '3 ) + 1 σ '2 +σ '3 )sin ϕ 2( (σ '3 −σ '2 ) + 1 σ '3 +σ '2 )sin ϕ 2( (σ '3 −σ '1 ) + 1 σ '3 +σ '1 )sin ϕ 2( (σ '1 −σ '3 ) + 1 σ '1 +σ '3 )sin ϕ 2( (σ '1 −σ '2 ) + 1 σ '1 +σ '2 )sin ϕ 2( (σ '2 −σ '1 ) + 1 σ '2 +σ '1 )sin ϕ 2( -σ1 − c cos ϕ ≤ 0 − c cos ϕ ≤ 0 − c cos ϕ ≤ 0 − c cos ϕ ≤ 0 − c cos ϕ ≤ 0 − c cos ϕ ≤ 0 (3. this condition ensures that Coulomb's friction law is obeyed in any plane within a material element.7f) -σ 3 -σ2 . The full Mohr-Coulomb yield condition consists of six yield functions when formulated in terms of principal stresses (see for instance Smith & Griffith.7e) (3. In fact.2 FORMULATION OF THE MOHR-COULOMB MODEL The Mohr-Coulomb yield condition is an extension of Coulomb's friction law to general states of stress.7a) (3.7c) (3.7b) (3.3.7d) (3. 9c) . using a sharp transition from one yield surface to another.8c) (3. i. the standard Mohr-Coulomb criterion allows for tension. the dilatancy angle ψ. The tension cut-off introduces three additional yield functions.2. 1982). however. defined as: f4 = σ1' – σt ≤ 0 f5 = σ2' – σt ≤ 0 f6 = σ3' – σt ≤ 0 (3.8b) (3.8e) (3.9a) (3. soil can sustain none or only very small tensile stresses. In addition to the yield functions.8d) (3. Some programs use a smooth transition from one yield surface to another.9b) (3. When implementing the Mohr-Coulomb model for general stress states. These yield functions together represent a hexagonal cone in principal stress space as shown in Figure 3. This behaviour can be included in a PLAXIS analysis by specifying a tension cut-off. In PLAXIS.The two plastic model parameters appearing in the yield functions are the well-known friction angle ϕ and the cohesion c.8f) The plastic potential functions contain a third plasticity parameter. 1990). van Langen & Vermeer. special treatment is required for the intersection of two yield surfaces. the exact form of the full Mohr-Coulomb model is implemented. allowable tensile stresses increase with cohesion. 1960.8a) (3.e. For c > 0. This parameter is required to model positive plastic volumetric strain increments (dilatancy) as actually observed for dense soils. Mohr circles with positive principal stresses are not allowed. In this case. In fact. For a detailed description of the corner treatment the reader is referred to the literature (Koiter. In reality. six plastic potential functions are defined for the Mohr-Coulomb model: g1a = g1b = g2a = g 2b = g 3a = g 3b = 1 2 1 2 1 2 1 2 1 2 1 2 (σ '2 −σ '3 ) + 1 σ '2 +σ '3 )sin ψ 2( (σ '3 −σ '2 ) + 1 σ '3 +σ '2 )sin ψ 2( (σ '3 −σ '1 ) + 1 σ '3 +σ '1 )sin ψ 2( (σ '1 −σ '3 ) + 1 σ '1 +σ '3 )sinψ 2( (σ '1 −σ '2 ) + 1 σ '1 +σ '2 )sinψ 2( (σ '2 −σ '1 ) + 1 σ '2 +σ '1 )sinψ 2( (3. A discussion of all of the model parameters used in the Mohr-Coulomb model is given at the end of this section. the rounding-off of the corners (see for example Smith & Griffith. Hence. which are generally familiar to most geotechnical engineers and which can be obtained from basic tests on soil samples. input is required on the elastic Young's modulus E and Poisson's ratio ν. For stress states within the yield surface. the allowable tensile stress. 3. as discussed in Section 2.3 BASIC PARAMETERS OF THE MOHR-COULOMB MODEL The Mohr-Coulomb model requires a total of five parameters. For these three yield functions an associated flow rule is adopted. σt. the behaviour is elastic and obeys Hooke's law for isotropic linear elasticity. besides the plasticity parameters c.When this tension cut-off procedure is used.2. is. and ψ.3 Parameter tab sheet for Mohr-Coulomb model . taken equal to zero. by default. These parameters with their standard units are listed below: E : : : : : Young's modulus Poisson's ratio Friction angle Cohesion Dilatancy angle [kN/m2] [-] [°] [kN/m2] [°] ν ϕ c ψ Figure 3. ϕ. as in the case of tunnelling and excavations. Hence. and the first loading modulus.4). but for loading of soils one generally uses E50. The stiffness is much higher for unloading and reloading than for primary loading.Young's modulus (E) PLAXIS uses the Young's modulus as the basic stiffness modulus in the elastic model and the Mohr-Coulomb model. it may be realistic to use such a . PLAXIS offers a special option for the input of a stiffness increasing with depth (see Section 3.σ 3| E0 1 E50 1 strain -ε1 Figure 3. when using a constant stiffness modulus to represent soil behaviour one should choose a value that is consistent with the stress level and the stress path development. both the unloading modulus. |σ1. consequently. E50. Considering unloading problems. In soil mechanics the initial slope is usually indicated as E0 and the secant modulus at 50% strength is denoted as E50 (see Figure 3. Note that some stressdependency of soil behaviour is taken into account in the advanced models in PLAXIS. such as particular unloading problems. For the Mohr-Coulomb model. one needs Eur instead of E50. the observed stiffness depends on the stress path that is followed. Poisson's ratio (ν) Standard drained triaxial tests may yield a significant rate of volume decrease at the very beginning of axial loading and. For materials with a large linear elastic range it is realistic to use E0. Moreover. Also. but some alternative stiffness moduli are displayed as well. the observed soil stiffness in terms of a Young's modulus may be lower for (drained) compression than for shearing. A stiffness modulus has the dimension of stress. a low initial value of Poisson's ratio (ν0). For some cases.4). tend to increase with the confining pressure. Hence. Eur. deep soil layers tend to have greater stiffness than shallow layers. which are described in Chapters 5 and 6. The values of the stiffness parameter adopted in a calculation require special attention as many geomaterials show a nonlinear behaviour from the very beginning of loading.4 Definition of E0 and E50 for standard drained triaxial test results For soils. ν is evaluated by matching K0. As both models will give the well-known ratio of σh / σν = ν / (1-ν) for one-dimensional compression it is easy to select a Poisson's ratio that gives a realistic value of K0.15 and 0. but some options will not perform well.4. it is more common to use values in the range between 0. will substantially increase plastic computational effort. PLAXIS offers a special option for the input of layers in which the cohesion increases with depth (see Section 3.25. In many cases one will obtain ν values in the range between 0. High friction angles.3 and 0. Friction angle (ϕ) The friction angle.4). shear stress Φ -σ1 -σ3 . such values can also be used for loading conditions other than one-dimensional compression. however.low initial value. In general. For this type of loading PLAXIS should give realistic ratios of K0 = σh / σν. Hence. Cohesion (c) The cohesive strength has the dimension of stress. ϕ (phi). one touches Coulomb's envelope The computing time increases more or less exponentially with the friction angle. PLAXIS can handle cohesionless sands (c = 0). which deals with initial stress distributions.σ2 c -σ3 -σ2 normal -σ1 stress Figure 3. as sometimes obtained for dense sands. is entered in degrees. high friction angles should be avoided when performing preliminary computations for a particular project. For unloading conditions. Hence.5 Stress circles at yield. nonexperienced users are advised to enter at least a small value (use c > 0. This subject is treated more extensively in Appendix A of the Reference Manual. The friction angle largely determines the shear strength as shown in . To avoid complications. but in general when using the Mohr-Coulomb model the use of a higher value is recommended.2 kPa). The selection of a Poisson's ratio is particularly simple when the elastic model or MohrCoulomb model is used for gravity loading (increasing ΣMweight from 0 to 1 in a plastic calculation). Figure 3.5 by means of Mohr's stress circles. A more general representation of the yield criterion is shown in Figure 3.2. The Mohr-Coulomb failure criterion proves to be better for describing soil behaviour than the Drucker-Prager approximation, as the latter failure surface tends to be highly inaccurate for axisymmetric configurations. Dilatancy angle (ψ) The dilatancy angle, ψ (psi), is specified in degrees. Apart from heavily overconsolidated layers, clay soils tend to show little dilatancy (ψ ≈ 0). The dilatancy of sand depends on both the density and on the friction angle. For quartz sands the order of magnitude is ψ ≈ ϕ - 30°. For ϕ-values of less than 30°, however, the angle of dilatancy is mostly zero. A small negative value for ψ is only realistic for extremely loose sands. For further information about the link between the friction angle and dilatancy, see Bolton (1986). 3.4 ADVANCED PARAMETERS OF THE MOHR-COULOMB MODEL When using the Mohr-Coulomb model, the <Advanced> button in the Parameters tab sheet may be clicked to enter some additional parameters for advanced modelling features. As a result, an additional window appears as shown in Figure 3.6. The advanced features comprise the increase of stiffness and cohesive strength with depth and the use of a tension cut-off. In fact, the latter option is used by default, but it may be deactivated here, if desired. Figure 3.6 Advanced Mohr-Coulomb parameters window Increase of stiffness (Eincrement ) In real soils, the stiffness depends significantly on the stress level, which means that the stiffness generally increases with depth. When using the Mohr-Coulomb model, the stiffness is a constant value. In order to account for the increase of the stiffness with depth the Eincrement-value may be used, which is the increase of the Young's modulus per unit of depth (expressed in the unit of stress per unit depth). At the level given by the yref parameter, the stiffness is equal to the reference Young's modulus, Eref, as entered in the Parameters tab sheet. The actual value of Young's modulus in the stress points is obtained from the reference value and Eincrement. Note that during calculations a stiffness increasing with depth does not change as a function of the stress state. Increase of cohesion (cincrement ) PLAXIS offers an advanced option for the input of clay layers in which the cohesion increases with depth. In order to account for the increase of the cohesion with depth the cincrement-value may be used, which is the increase of cohesion per unit of depth (expressed in the unit of stress per unit depth). At the level given by the yref parameter, the cohesion is equal to the (reference) cohesion, cref, as entered in the Parameters tab sheet. The actual value of cohesion in the stress points is obtained from the reference value and cincrement. Tension cut-off In some practical problems an area with tensile stresses may develop. According to the Coulomb envelope shown in Figure 3.5 this is allowed when the shear stress (radius of Mohr circle) is sufficiently small. However, the soil surface near a trench in clay sometimes shows tensile cracks. This indicates that soil may also fail in tension instead of in shear. Such behaviour can be included in PLAXIS analysis by selecting the tension cut-off. In this case Mohr circles with positive principal stresses are not allowed. When selecting the tension cut-off the allowable tensile strength may be entered. For the Mohr-Coulomb model and the Hardening-Soil model the tension cut-off is, by default, selected with a tensile strength of zero. These joint sets have to be parallel. G2 Parameters σt. as considered in the Jointed Rock model. distinction can be made between elastic anisotropy and plastic anisotropy. the tensile stresses perpendicular to the three planes are limited according to a predefined tensile strength (tension cut-off). i Parameters ci. Elastic anisotropy refers to the use of different elastic stiffness properties in different directions. Upon reaching the maximum shear stress in such a direction. they may respond differently when subjected to particular conditions in one direction or another. The intact rock is considered to behave as a transversly anisotropic elastic material.4 THE JOINTED ROCK MODEL (ANISOTROPY) Materials may have different properties in different directions. E2. especially meant to simulate the behaviour of stratified and jointed rock layers. Another form of plastic anisotropy is kinematic hardening. not filled with fault gouge.i • Shear failure according to Coulomb in three directions. As a result. i Parameters E1. This aspect of material behaviour is called anisotropy. quantified by five parameters and a direction. ν2. The application of the Jointed Rock model is justified when families of joints or joint sets are present. In this model it is assumed that there is intact rock with an eventual stratification direction and major joint directions. Each plane may have different shear strength properties. In the major joint directions it is assumed that shear stresses are limited according to Coulomb's criterion. plastic sliding will occur. A maximum of three sliding directions ('planes') can be defined. rock formation stratification major joint direction Figure 4. The latter is not considered in PLAXIS program. and their spacing has to be small compared to the characteristic dimension of the structure.1 Visualization of concept behind the Jointed Rock model The Jointed Rock model is an anisotropic elastic perfectly-plastic model. In addition to plastic shearing. of which the first plane is assumed to coincide with the direction of elastic anisotropy. ν1. Plastic anisotropy may involve the use of different strength properties in different directions. When modelling anisotropy. ϕi and ψi . The anisotropy may result from stratification or from other phenomena. Some basic characteristics of the Jointed Rock model are: • Anisotropic elastic behaviour for intact rock • Limited tensile strength in three directions. where the stiffness in horizontal direction.4. the D*-matrix as used in the Jointed Rock model is transversely anisotropic. but the above relations will generally hold for a local (n. the stratification plane will not be parallel to the global x-z-plane. a horizontal stratification. This direction may correspond to the stratification direction or to any other direction with significantly different elastic stiffness properties. As a consequence. In general.1e) (4. E2. Therefore we consider first a transformation of stresses and strains: .s.1c) γ! xy = γ! yz = !zx = γ ! xy σ G2 ! yz σ G2 ! zx 2(1 + ν1 )σ E1 (4.1a) ! yy = − ε ! zz = − ε (4.t) coordinate system where the stratification plane is parallel to the s-t-plane. (D*)-1. In this case the 'Plane 1' direction is parallel to the x-z-plane and the following constitutive relations exist (See: Zienkiewicz & Taylor: The Finite Element Method. 4th Ed.1 ANISOTROPIC ELASTIC MATERIAL STIFFNESS MATRIX The elastic material behaviour in the Jointed Rock model is described by an elastic material stiffness matrix. In contrast to Hooke's law. This matrix is symmetric. The regular material stiffness matrix D* can only be obtained by numerical inversion.1f) The inverse of the anisotropic elastic material stiffness matrix.): !xx = ε ! yy ν 1σ ! xx ν 2σ ! zz σ − − E1 E2 E1 ! yy ν 2σ σ ! xx ! zz ν 2σ + − E2 E2 E2 ! yy ! xx ν 2σ ! ν 1σ σ − + zz E1 E2 E1 (4.3). is different from the stiffness in vertical direction. the local material stiffness matrix has to be transformed from the local to the global coordinate system. E1. follows from the above relations. The orientation of this plane is defined by the dip angle and dip direction (see 4. D*. Consider. Different stiffnesses can be used normal to and in a predefined direction ('plane 1').1d) (4. for example.1b) (4. σ nst = Rσ σ xyz ε nst = Rε ε xyz where Rσ = nx sx nx nx sx tx 2 2 2 σ xyz = Rσ σ nst ε xyz = Rε ε nst nz sz tz 2 2 2 -1 (4.y.2b) 2 nx n z 2 sx sz 2 tx tz n z s x + nx s z sx t z + sz t x nz t x+ nx t z n y nz s y sz ty tz -1 ny sy ty 2 2 2 2 nx ny 2 sx s y 2 tx ty 2 n y nz 2 s y sz 2 t y tz (4.z)-coordinates (i. sx. sz. sy.5) A local stress-strain relationship in (n. tx. ty and tz are the components of the normalized n.y.3).4) n z s x + nx s z sx t z + sz t x n z t x + nx t z tx tz s x 2 n y s y 2 n z s z n x s y + n y s x n y s z + nz s y tx tx 2 sy t y 2 ny t y 2 sz t z 2 nz t z sx t y+s y t x nx t y + n y t x s y t z + sz t y n y t z + nz t y nx.z)-coordinates in the following way: σ nst = D* ε nst nst σ nst = R σ σ xyz ε nst = R ε ε xyz ⇒ R σ σ xyz = D nst R ε ε xyz * (4.e.3) s x n y s y nz s z n x s y + n y s x n y s z + n z s y tx tx 2 2 2 sy ty ny t y sz t z nz t z 2 2 2 sx t y+ s y t x nx t y + n y t x 2 2 2 s y t z + sz t y n y t z + nz t y nx n y sx s y tx ty and Rε = 2 nx 2 sx 2 nx nx sx ny sy ty nz sz nx n z sx sz t x t z (4. ny. s and t-vectors in global (x.s.6) . 'sines' and 'cosines'.t)-coordinates can be transformed to a global relationship in (x. For plane condition n z = s z = t x = t y = 0 and t z = 1 . nz. see 4.2a) (4. It further holds that : R ε = Rσ T -1 Rσ = R ε T -1 (4. τ. In addition. σn.e. ε nst = D nst σ nst σ nst = Rσ σ xyz ε nst = R ε ε xyz Hence. i. Each plane. Moreover. On each plane a local Coulomb condition applies to limit the shear stress.φ i .i . after which the total is numerically inverted to obtain the global material stiffness matrix D*xyz. τs and τt.9) D xyz * -1 = Rσ Dnst T * -1 Rσ or T * -1 * D xyz = Rσ D nst Rσ -1 (4.ψ i and σt.t)-coordinates it is necessary to calculate the local stresses from the Cartesian stresses. a normal stress component. a tension cut-off criterion is used to limit the tensile stress on a plane. i.11) . However. The local stresses involve three components. σ i =TT σ i (4. has its own strength parameters ci . a maximum of two other sliding directions may be defined.5): T * * σ xyz = R ε D nst R ε ε xyz = D xyz ε xyz or * * D xyz = R ε D nst R ε T (4.Hence.7) Using to above condition (4.8) Actually. The first sliding plane corresponds to the direction of elastic anisotropy. 4. and two independent shear stress components. not the D*-matrix is given in local coordinates but the inverse matrix (D*)-1.s. the formulation of plasticity on all planes is similar. * σ xyz = Rσ D nst R ε ε xyz 1 (4. * −1 -1 * -1 T * -1 ⇒ ε xyz = Rε Dnst Rσ σ xyz = Rσ Dnst Rσ σ xyz (4. the transformation is considered first.10) Instead of inverting the (D*nst)-1-matrix in the first place.2 PLASTIC BEHAVIOUR IN THREE DIRECTIONS A maximum of 3 sliding directions (sliding planes) can be defined in the Jointed Rock model. In order to check the plasticity conditions for a plane with local (n. for plane i T As usual in PLAXIS. tensile (normal) stresses are defined as positive whereas compression is defined as negative. Here a sliding plane is considered under an angle α1 (= dip angle) with respect to the x-axis.2 Plane strain situation with a single sliding plane and vectors n.12a) (4.2.12b) σ = (σ xx σ yy σ zz σ xy σ yz σ zx )T T i = transformation matrix (3x6).13) s = sin α1 c = cos α1 .where σ i = (σ n τs τt ) T (4. s Consider a plane strain situation as visualized in Figure 4. y s n α1 α1 sliding plane x Figure 4. In this case the transformation matrix TT becomes: -2sc 0 0 s2 c2 0 T T = sc -sc 0 -s2 + c2 0 0 0 0 0 0 -c -s where (4. 16) . since it involves both the dip angle and the dip direction (see 4.3 visualizes the full yield criterion on a single plane. The yield functions for plane i are defined as: f i = τ s + σ n tan φi − ci t f i = σ n − σ t . After having determined the local stress components. the plasticity conditions can be checked on the basis of yield functions.i (Coulomb) (Tension cut-off) (4. 4 and 6 of Rσ (see Eq. TT.3 Yield criterion for individual plane The local plastic strains are defined by: ∆ε p j =λj ∂gj ∂σ j (4.In the general three-dimensional case the transformation matrix is more complex.15a) (4.i ci -σ n Figure 4.3): 2 2 2 2 nx ny 2 ny nz 2 nz nx nx ny nz T T = nx sx ny s y nz sz nx s y+ny sx nz s y +ny sz nz sx+nx sz nx t x ny t y nz t z ny t x+nx t y ny t z +nz t y nz t x+nx t z (4.15b) ( σ t .3). for the calculation of local stresses corresponds to rows 1.i ≤ ci cot φi ) Figure 4. 4. |τ| ϕi σt.14) Note that the general transformation matrix. is also used to transform the local plastic strain increments of plane j.3 PARAMETERS OF THE JOINTED ROCK MODEL Most parameters of the jointed rock model coincide with those of the isotropic MohrCoulomb model. j (Coulomb) (Tension cut-off) (4.3): E1 : : Young's modulus for rock as a continuum Poisson's ratio for rock as a continuum [kN/m2] [ −] ν1 . there are 26 = 64 possibilities of (combined) yielding. so up to 6 plastic multipliers must be found such that all yield functions are at most zero and the plastic multipliers are non-negative. a maximum of 6 yield functions exist. T.17a) (4. 4. into global plastic strain increments.19b) Ti D T j ∂σ ∂σ T This means finding up to 6 values of λi ≥ 0 such that all fi ≤ 0 and λi fi = 0 When the maximum of 3 planes are used. ∆ε p: ∆ε p = T j ∆ε p j (4.18) The consistency condition requires that at yielding the value of the yield function must remain zero for all active yield functions. all these possibilities are taken into account in order to provide an exact calculation of stresses. In the calculation process.17b) The transformation matrix.19a) <λtj > − Ti D T j Ti D T j ∂σ ∂σ j=1 ∂σ ∂σ T ∑ np T i ft = ie ft − ∑ j=1 np < λcj > ∂ f it T ∂ g cj − Ti D T j ∂σ ∂σ T ∑ j=1 < λtj > ∂ f it T ∂ gtj (4. i fc = ie fc − ∑ j=1 np <λcj > ∂ f ic T ∂ gcj np ∂ f ic T ∂ gtj (4.where gj is the local plastic potential function for plane j: g j = τ j + σ n tan φ j − c j g tj = σ n − σ t . For all planes together. Elastic parameters as in Mohr-Coulomb model (see Section 3. These are the basic elastic parameters and the basic strength parameters. ∆ε pj. 3): ci : : : : Cohesion Friction angle Dilatancy angle Tensile strength [kN/m2] [°] [°] [kN/m2] ϕi ψ σt.4 Parameters for the Jointed Rock model . stratification direction): E2 G2 : : : Young's modulus in 'Plane 1' direction Shear modulus in 'Plane 1' direction Poisson's ratio in 'Plane 1' direction [kN/m2] [kN/m2] [− ] ν2 Strength parameters in joint directions (Plane i=1. 2. 3): n : : : Numer of joint directions (1 ≤ n ≤ 3) Dip angle Dip direction [°] [°] α1.i α2.i Definition of joint directions (Plane i=1. 2.i Figure 4.Anisotropic elastic parameters 'Plane 1' direction (e.g. The dilatancy angle. The strength properties ci and ϕi determine the allowable shear strength according to Coulomb's criterion and σt determines the tensile strength according to the tension cut-off criterion. In general. These directions may correspond to the most critical directions of joints in the rock formation. The latter is displayed after pressing <Advanced> button. G2 is a separate parameter and is not simply related to Young's modulus by means of Poisson's ratio (see Eq.1d and e). i. together with a second Poisson's ratio. as if it would not be anisotropic. In general. ψi. the number of sliding planes (= sliding directions) is still 1. 4. Strength parameters Each sliding direction (plane) has its own strength properties ci. Elastic anisotropy in a rock formation may be introduced by stratification. the tension cut-off is active and the tensile strength is set to zero. and determines the plastic volume expansion due to shearing. ν2. In contrast to Hooke's law of isotropic elasticity. Definition of joint directions It is assumed that the direction of elastic anisotropy corresponds with the first direction where plastic shearing may occur ('Plane 1'). it is also common in geology to use the Strike.Elastic parameters The elastic parameters E1 and ν1 are the (constant) stiffness (Young's modulus) and Poisson's ratio of the rock as a continuum according to Hooke's law. and therefore the unambiguous Dip direction as mostly used by rock engineers is used in PLAXIS. By default. and strength parameters must be specified for this direction anyway. Elastic shearing in the stratification direction is also considered to be 'weaker' than elastic shearing in other directions. ϕi and σt. then the parameters E2 and ν2 can be simply set equal to E1 and ν1 respectively. However. care should be taken with the definition of Strike. is used in the plastic potential function g. A maximum of three sliding directions can be defined. This direction must always be specified. In the case the rock formation is stratified without major joints. This reduced stiffness can be represented by the parameter E2. The stiffness perpendicular to the stratification direction is usually reduced compared with the general stiffness. The sliding directions are defined by means of two parameters: The Dip angle (α1) (or shortly Dip) and the Dip direction (α2). If the elastic behaviour of the rock is fully isotropic. .5.i and dilatancy angle ψi. whereas G2 should be set to ½E1/(1+ν1). the shear stiffness in the anisotropic direction can explicitly be defined by means of the elastic shear modulus G2. The definition of both parameters is visualized in Figure 4. the elastic stiffness normal to the direction of elastic anisotropy is defined by the parameters E2 and ν2.e. Instead of the latter parameter. In addition to the orientation of the sliding planes it is also known how the global (x. The dip direction is entered in the range [0°. as defined in the General settings in the Input program.z) model coordinates relate to the North direction. The orientation of the sliding plane is further defined by the dip direction.y N t α2 s* α1 s α1 sliding plane n Figure 4. which are both normal to the vertical y-axis.5. The angle α1 is the dip angle. Hence. The sliding plane can be defined by the vectors (s. whereas the vector s is the 'fall line' of the sliding plane and the vector t is the 'horizontal line' of the sliding plane.t). . The sliding plane makes an angle α1 with respect to the horizontal plane.y.5 Definition of dip angle and dip direction Consider a sliding plane. which is the orientation of the vector s* with respect to the North direction (N). where the horizontal plane can be defined by the vectors (s*. α2.t). The Declination is the positive angle from the North direction to the positive z-direction of the model. 360°]. measured clockwise to the horizontal projection of the fall line (=s*-direction) when looking downwards. as indicated in Figure 4. which is defined as the positive 'downward' inclination angle between the horizontal plane and the sliding plane. α1 is the angle between the vectors s* and s. This information is contained in the Declination parameter. The dip direction is defined as the positive angle from the North direction. which are both normal to the vector n. The vector n is the 'normal' to the sliding plane. The dip angle must be entered in the range [0°. 90°]. measured clockwise from s* to s when looking in the positive t-direction. 20c) Below some examples are shown of how sliding planes occur in a 3D models for different values of α1.6 Definition of various directions and angles in the horiziontal plane In order to transform the local (n. α3 is defined as the positive angle from the positive z-direction clockwise to the s*-direction when looking downwards.s. From the definitions as given above. being the difference between the Dip direction and the Declination: α3 = α2 − Declination (4.z) coordinate system. α2 and Declination: .y α2 s* α3 z N x declination Figure 4. it follows that: n x − sin α1 sin α 3 cos α1 n= n y = α α sin cos n 1 3 z s x − cos α1 sin α 3 s= s y = − sin α1 sz cos α1 cos α 3 t x cos α 3 t = t y = 0 t z sin α 3 (4.19) Hence.20a) (4.t) coordinate system into the global (x. an auxiliary angle α3 is used internally.y.20b) (4. for plane strain conditions (the cases considered in Version 8) only α1 is required. α2 is fixed at 90º and the declination is set to 0º. α2 and Declination As it can be seen. .7 Examples of failure directions defined by α1 . By default.y α1 = 45º α2 = 0º Declination = 0º z y x α1 = 45º α2 = 90º Declination = 0º z y x α1 = 45º α2 = 0º Declination = 90º z x Figure 4. The Hardening-Soil model is an advanced model for simulating the behaviour of different types of soil. In the special case of a drained triaxial test. The Hardening-Soil model.7) ( ) m ref E oed = p ref λ∗ λ∗ = λ (1 + e0 ) where pref is a reference pressure. supersedes the hyperbolic model by far. Distinction can be made between two main types of hardening. however. both soft soils and stiff soils. Hence. Elastic unloading / reloading. soil shows a decreasing stiffness and simultaneously irreversible plastic strains develop. the observed relationship between the axial strain and the deviatoric stress can be well approximated by a hyperbola. Failure according to the Mohr-Coulomb model. Shear hardening is used to model irreversible strains due to primary deviatoric loading. Schanz (1998). νur Parameters c. Such a relationship was first formulated by Kondner (1963) and later used in the well-known hyperbolic model (Duncan & Chang. as used in the the Soft-Soil model and the oedometer loading modulus (see also Section 6. the yield surface of a hardening plasticity model is not fixed in principal stress space. In the special case of soft soils it is realistic to use m = 1. In such situations there is also a simple relationship between the modified compression index λ*. Some basic characteristics of the model are: • • • • • Stress dependent stiffness according to a power law. Plastic straining due to primary compression. Here we consider a tangent oedometer modulus at a particular reference pressure pref. Compression hardening is used to model irreversible plastic strains due to primary compression in oedometer loading and isotropic loading. namely shear hardening and compression hardening. When subjected to primary deviatoric loading. 1970).5 THE HARDENING-SOIL MODEL (ISOTROPIC HARDENING) In contrast to an elastic perfectly-plastic model. Plastic straining due to primary deviatoric loading. For oedometer conditions of stress and strain. ϕ and ψ A basic feature of the present Hardening-Soil model is the stress dependency of soil stiffness. but it can expand due to plastic straining. Both types of hardening are contained in the present model. the model implies for example ref the relationship E oed = E oed σ / pref . Secondly by including soil dilatancy and thirdly by introducing a yield cap. the primary loading stiffness relates to the modified compression index λ*. Input parameter m ref Input parameter E 50 ref Input parameter E oed ref Input parameters E ur . . Firstly by using the theory of plasticity rather than the theory of elasticity. and the deviatoric stress. Janbu (1963) reports values of m around 0.1 HYPERBOLIC RELATIONSHIP TRIAXIAL TEST FOR STANDARD DRAINED A basic idea for the formulation of the Hardening-Soil model is the hyperbolic relationship between the vertical strain.Similarly. (5.1) Where qa is the asymptotic value of the shear strength. The actual stiffness depends on the minor principal stress. There is the approximate relationship: ref E ur = 3 p ref (1 − 2νur ) κ∗ κ∗ = κ (1 + e0 ) Again. σ3'. In PLAXIS. Here standard drained triaxial tests tend to yield curves that can be described by: − ε1 = 1 q 2 E50 1 − q / q a for : q < q f (5. The parameter E50 is the confining stress dependent stiffness modulus for primary loading and is given by the equation: E50 = ref ref E 50 c cos ϕ − σ 3 ′ sin ϕ c cos ϕ + p ref sin ϕ m (5. 5.2) where E 50 is a reference stiffness modulus corresponding to the reference confining pressure pref. qf.3) .5 for Norwegian sands and silts. the unloading-reloading modulus relates to the modified swelling index κ*.5 < m < 1. The ultimate deviatoric stress. the power should be taken equal to 1. in primary triaxial loading. this relationship applies in combination with the input value m = 1. a default setting pref = 100 stress units is used. ε1.1) are defined as: ′) q f = (c cot ϕ − σ 3 2 sin ϕ 1 − sin ϕ and : qa = qf Rf (5. Please note that σ'3 is negative for compression.0. as observed for soft clays. which is the confining pressure in a triaxial test. This relationship is plotted in Figure 5. The amount of stress dependency is given by the power m. q. In order to simulate a logarithmic stress dependency. and the quantity qa in Eq.1. whilst Von Soos (1980) reports various different values in the range 0.0. The ratio between qf and qa is given by the failure ratio Rf. Rf = 0.1. As soon as q = qf .2 APPROXIMATION OF HYPERBOLA BY THE HARDENING-SOIL MODEL For the sake of convenience. deviatoric stress |σ1-σ3| qa qf E50 1 Eur 1 asymptote failure line ref axial strain -ε1 Figure 5. another stress-dependent stiffness modulus is used: Eur = ref ref E ur c cos ϕ − σ 3 ′ sin ϕ c cos ϕ + p ref sin ϕ m (5. corresponding to the reference pressure pref. which involves the strength parameters c and ϕ. The above relationship for qf is derived from the Mohr-Coulomb failure criterion. It should also be realised that compressive stress and strain are considered positive.1 Hyperbolic stress-strain relation in primary loading for a standard drained triaxial test 5.9 is chosen as a suitable default setting. In PLAXIS. For unloading and reloading stress paths. which should obviously be smaller than 1. Moreover. For a more general presentation of the Hardening-Soil . this is the default setting used in PLAXIS. the failure criterion is satisfied and perfectly plastic yielding occurs as described by the Mohr-Coulomb model.4) where E ur is the reference Young's modulus for unloading and reloading.Again it is remarked that σ'3 is usually negative. restriction is again made to triaxial loading conditions with σ2' = σ3' and σ1' being the major compressive stress. it is assumed that q < qf. In many practical cases it is appropriate to set E ur equal ref to 3 E 50 . as also indicated in Figure 5. The above definition of the strain-hardening parameter γ p will be referred to later.1). one has to consider primary loading.8) where νur is the unloading / reloading Poisson's ratio. p as this implies the yield condition f = 0. the Hardening-Soil model predicts fully elastic volume changes according to Hooke's law. For checking this statement. Plastic strains develop in primary loading alone. An essential feature of the above definitions for f is that it matches the well-known hyperbolic law (5. it thus yields γ = f and it follows from Eqs. (5. For drained triaxial test stress paths with σ2' = σ3' = constant.6) with q.model the reader is referred to Schanz et al (1999). whilst the strains that develop during the very first stage of the test are not considered. Here it should be realised that restriction is made to strains that develop during deviatoric loading. For primary loading. .4). This stems from a yield function of the form: f = f −γ p where f is a function of stress and γ p is a function of plastic strains: (5. Let us first consider the corresponding plastic strains. (5. but these strains are not included in Eq. whilst the superscript p is used to denote plastic strains.8). qa. For hard soils. (5. In this section it will be shown that this model gives virtually the hyperbolic stress strain curve of Eq. For the first stage of isotropic compression (with consolidation).7) In addition to the plastic strains. the elastic Young's modulus Eur remains constant and the elastic strains are given by the equations: e − ε1 = q E ur e −εe 2 = − ε 3 = − ν ur q E ur (5.6) that: − ε 1p ≈ 1 2 p f = 1 q q − 2 E50 1 − q / q a E ur (5.2) to (5. but elastic strains develop both in primary loading and unloading / reloading. the model accounts for elastic strains. plastic volume changes ( ε v ) tend to be relatively small and this leads to the approximation γ p ≈ −2ε1p . (5.1) when considering stress paths of standard drained triaxial tests.5) f = 2q q − E 50 1 − q / q a E ur 1 γ p = − 2ε 1p − ε vp ≈ −2ε 1p ( ) (5. E50 and Eur as defined by Eqs. (5. plastic volumetric strains will never be precisely equal to zero. For m = 1. the axial strain is the sum of an elastic component given by Eq. when ε vp = 0.7). can be visualised in p'-q-plane by means of a yield locus.e.2 Successive yield loci for various constant values of the hardening parameter γ p .2) and (5. it follows that: − ε 1 = −ε 1e − ε 1p ≈ 1 q 2 E50 1 − q / qa (5. (5. Hence. being typical for hard soils. γ p. but slightly curved yield loci correspond to lower values of the exponent. the yield condition f = 0.2 shows the shape of successive yield loci for m = 0. For a given constant value of the hardening parameter. (5.For the deviatoric loading stage of the triaxial test. In reality. Because of the latter expressions. deviatoric stress |σ 1-σ3| Mohr-Coulomb failure line Mean effective stress Figure 5. straight lines are obtained.6) as well as Eqs. When plotting such yield loci. so that the approximation in Eq.5. It is thus made clear that the present Hardening-Soil model yields a hyperbolic stress-strain curve under triaxial testing conditions. but for hard soils plastic volume changes tend to be small when compared with the axial strain. the shape of the yield loci depend on the exponent m. i. (5.9) This relationship holds exactly in absence of plastic volume strains. (5. Figure 5.8) and a plastic component according to Eq.4) for E50 and Eur respectively.9) will generally be accurate. one has to use Eq. and ϕm is the mobilised friction angle: sinϕ m = ′ −σ3 ′ σ1 ′ +σ3 ′ − 2 c cot ϕ σ1 (5. it is found from Eq. where ϕcν is the critical state friction angle. being a material constant independent of density. ψ. ϕ.11) is adopted. whilst dilatancy occurs for high stress ratios (ϕm > ϕcν). PLAXIS performs this computation automatically and so users do not need to specify a value for ϕcν. the expression: sin ψ m = sin ϕ m − sin ϕ cv 1 − sin ϕ m sin ϕ cv (5. . The essential property of the stressdilatancy theory is that the material contracts for small stress ratios (ϕm < ϕcν).13b) Hence. the critical state angle can be computed from the failure angles ϕ and ψ.3 PLASTIC VOLUMETRIC STRAIN FOR TRIAXIAL STATES OF STRESS Having presented a relationship for the plastic shear strain. (5.10) Clearly. Instead.13a) sin ϕ cv = sin ϕ − sin ψ 1 − sin ϕ sin ψ (5. further detail is needed by specifying the mobilised dilatancy angle ψm. ε v . ϕ. a relationship between p and γ ! p . At failure. the Hardening-Soil model involves a relationship between rates of plastic strain. γ p. This flow rule has the linear form: !v ε p !vp = sinψm γ! ε (5. As for all plasticity models. when the mobilised friction angle equals the failure angle. For the present model. as explained by Schanz & Vermeer (1995). and the ultimate dilatancy angle.e. i. attention is now focused p on the plastic volumetric strain.5.11) that: sin ψ = or equivalently: sin ϕ − sin ϕ cv 1 − sin ϕ sin ϕ cv (5. one has to provide input data on the ultimate friction angle.12) The above equations correspond to the well-known stress-dilatancy theory by Rowe (1962). 5.4 PARAMETERS OF THE HARDENING-SOIL MODEL Some parameters of the present hardening model coincide with those of the nonhardening Mohr-Coulomb model.3 Basic parameters for the Hardening-Soil model Failure parameters as in Mohr-Coulomb model (see Section 3. ϕ and ψ. Figure 5. These are the failure parameters c.3): c : (Effective) cohesion : (Effective) angle of internal friction : Angle of dilatancy [kN/m2] [°] [°] ϕ ψ Basic parameters for soil stiffness: E 50 ref : Secant stiffness in standard drained triaxial test : Tangent stiffness for primary oedometer loading : Power for stress-level dependency of stiffness [kN/m2] [kN/m2] [-] E oed m ref Advanced parameters (it is advised to use the default setting): E ur νur ref ref ref : Unloading / reloading stiffness (default E ur = 3 E 50 ) [kN/m2] [-] : Poisson's ratio for unloading-reloading (default νur = 0.2) . Please note that PLAXIS allows for the input of Eur and νur but not for a direct input of Gur. E oed tangent stiffness at a vertical stress of −σ'1 = pref. Hence. As some PLAXIS users are familiar with the input of shear moduli rather than the above stiffness moduli. there is no simple conversion from E50 to G50.2). a ref ′ = p ref stiffness modulus E 50 is defined for a reference minor principal stress of − σ 3 As a default value. Here we use the equation: E oed = ref E oed c cos ϕ − σ 3 ′ sin ϕ c cos ϕ + p ref sin ϕ m (5.9) (see Figure 5.1) : Tensile strength (default σtension = 0 stress units) σtension cincrement : As in Mohr-Coulomb model (default cincrement = 0) ref ref & Eoed and power m Stiffness moduli E50 The advantage of the Hardening-Soil model over the Mohr-Coulomb model is not only the use of a hyperbolic stress-strain curve instead of a bi-linear curve. Instead.4. where Gur is an elastic shear modulus. . Having defined E50 by Eq. It is therefore necessary to estimate the stress levels within the soil and use these to obtain suitable values of stiffness. shear moduli will now be discussed. one may thus write Eur = 2 (1+ν) Gur. With the Hardening-Soil model. As Eur is a real elastic stiffness. (5. the program uses pref = 100 stress units. these stiffnesses can be inputted independently. the user has to select a fixed value of Young's modulus whereas for real soils this stiffness depends on the stress level. it is now important to define the oedometer stiffness. Note that we use σ1 rather than σ3 and that we consider primary loading. Instead. but also the control of stress level dependency. In contrast to elasticity based models. the elastoplastic Hardening-Soil model does not involve a fixed relationship between the (drained) triaxial stiffness E50 and the oedometer stiffness Eoed for one-dimensional compression. Within Hooke's theory of elasticity conversion between E and G goes by the equation E = 2 (1+ν) G. however. the secant modulus E50 is not used within a concept of elasticity. As a consequence.14) ref is a where Eoed is a tangent stiffness modulus as indicated in Figure 5.pref : Reference stress for stiffnesses (default pref = 100 stress units) [kN/m2] nc : K0-value for normal consolidation (default K 0 = 1-sinϕ) K0 Rf nc [-] [-] [kN/m2] [kN/m3] : Failure ratio qf / qa (default Rf = 0. When using the Mohr-Coulomb model. In contrast to Eur. this cumbersome selection of input parameters is not required. Figure 5.4 Definition of E oed Advanced parameters Realistic values of νur are about 0.2 and this value is thus used as a default setting.5.-σ 1 E ref oed p ref 1 -ε1 ref in oedometer test results Figure 5. as indicated in Figure 5.5 Advanced parameters window . dilating materials arrive in a state of critical density where dilatancy has come to an end. and the maximum void ratio. is automatically set back to zero. as indicated in Figure 5.nc is not simply a function of Poisson's ratio. In order to specify this behaviour.15b) ψ = 0 . It is suggested to maintain this value as the correlation is highly realistic.6 Resulting strain curve for a standard drained triaxial test when including dilatancy cut-off for e < emax: sinψmob = sin ϕ mob − sin ϕ cv 1 − sin ϕ mob sin ϕ cv where: for e ≥ e : sinϕcν = sin ϕ − sin ψ 1 − sin ϕ sin ψ (5. This phenomenon of soil behaviour can be included in the Hardening-Soil model by means of a dilatancy cut-off. However. All possible different nc cannot be accommodated for. dilatancy cut-off OFF εv dilatancy cut-off ON maximum porosity reached 1 – sinψ 2 sinψ ε1 Figure 5. emax. ψmob. as indicated in Figure 5. the program shows the nearest possible value that will be used in the computations. the initial void ratio. As a default setting PLAXIS uses the correlation nc K 0 = 1−sinϕ. einit.6. In contrast to the Mohr-Coulomb model. users do have the possibility to select different values.15a) (5. K 0 but a proper input parameter. there happens to be a certain range of valid K 0 nc K 0 values outside this range are rejected by PLAXIS. Eoed. the mobilised dilatancy angle. input values for K 0 nc -values. Dilatancy cut-off After extensive shearing. of the material must be entered as general parameters. On inputting values.6. such as E50. Depending on other parameters. As soon as the volume change results in a state of maximum void. Eur and νur. of a soil can also be inputted. einit. E oed is used to control the magnitude of plastic strains that originate . Similarly. Figure 5. As soon as the maximum void ratio is reached. E ref 50 largely controls the magnitude of the plastic strains that are associated with the shear yield ref surface. but this general soil parameter is not used within the context of the Hardening-Soil model. is the in-situ void ratio of the soil body.The void ratio is related to the volumetric strain. The triaxial modulus largely controls the shear yield surface and the oedometer modulus controls the cap yield surface. emin. The maximum void ratio is the void ratio of the material in a state of critical void (critical state). The minimum void ratio. In fact. A second type of yield surface must therefore be introduced to close the elastic region in the direction of the p-axis. By default. Please note that the selection of the dilatancy cut-off and the input of void ratios is done in the 'general' tab sheet of the material data set window and not in the 'parameters' tab sheet.16) The initial void ratio. Without such a cap type yield surface it would not be possible to formulate a model with independent ref ref and input of both E 50 E oed . the dilatancy cut-off is not active. ( ) 1+ e (5. εν by the relationship: − ε v − ε init = ln v 1+ einit where an increment of εv is positive for dilatancy.5 ON THE CAP YIELD SURFACE IN THE HARDENING-SOIL MODEL Shear yield surfaces as indicated in Figure 5.7 Advanced general properties window 5. The selection of the dilatancy cut-off is only available when the Hardening-Soil model has been selected.2 do not explain the plastic volume strain that is measured in isotropic compression. the dilatancy angle is set to zero. In this section the yield cap will be described in full detail. q ~ = −(σ −σ ) and for triaxial extension triaxial compression (−σ1>−σ2=−σ3) it yields q 1 3 ~ reduces to q ~ = −δ (σ −σ ). as indicated in Figure 5. The ellipse is used both as a yield surface and as a plastic potential.from the yield cap. but we will not use them as direct input parameters. where α is an auxiliary model parameter that relates to K 0 ~ Further more we have p = − (σ1+σ2+σ3) /3 and q = σ1+(δ−1)σ2−δσ3 with δ=(3+sinϕ ) / ~ is a special stress measure for deviatoric stresses. In addition to the well known constants m and pref there is another model constant β. High values of α lead to steep caps underneath the Mohr-Coulomb line. For understanding the shape of the yield cap. Instead. p determines its The ellipse has length pp on the p-axis and α pp on the q p magnitude and α its aspect ratio. all be realised that it is an ellipse in p.18) The volumetric cap strain is the plastic volumetric strain in isotropic compression. Hence: ! ε pc ∂ fc = λ ∂σ with: pp β ref λ= 2 p p m !p p p ref (5. We have a hardening law relating pp to volumetric cap strain ε vpc : ε pc v pp β ref = 1− m p 1− m (5. Hence. it should first of ~ -plane.q ~ -axis. we have relationships of the form: nc α ↔ K0 ref β ↔ E oed nc = 1 − sin ϕ ) (default: K 0 ref ref (default: E oed = E 50 ) nc ref and E oed can be used as input parameters that determine the magnitude such that K 0 of α and β respectively.19) . whereas small α-values define caps that are much more pointed around the p-axis.17) nc as will be discussed later. To this end we consider the definition of the cap yield surface: fc = ~2 q 2 2 + p − pp 2 α (5. The magnitude of the yield cap is ( − σ =− σ >− σ ) q 1 2 3 1 3 determined by the isotropic pre-consolidation stress pp. Both α and β are cap parameters.8. In the special case of (3−sinϕ ). q can be further reduced by means of a tension cut-off -σ1 -σ 3 -σ2 Figure 5. pp is either computed from the entered overconsolidation ratio (OCR) or the pre-overburden pressure (POP) (see Section 2. The cap yield surface expands as a function of the pre-consolidation stress p .This expression for λ derives from the yield condition f c = 0 and Eq.9 Representation of total yield contour of the Hardening-Soil model in principal stress space for cohesionless soil The first figure shows simple yield lines. ~ q αpp elastic region p c cot φ pp ~ -plane. For understanding the yield surfaces in full detail.9. whereas the second one depicts yield surfaces in principal stress space.6). the shear yield locus can expand up to the ultimate Mohr-Coulomb failure surface. Here.8 Yield surfaces of Hardening-Soil model in p. The elastic region Figure 5. (5. Both the shear locus and the yield cap have the hexagonal shape of the classical Mohr-Coulomb failure criterion.8 and Figure 5. In fact.18) for pp. one should consider both Figure 5. . Input data on initial pp-values is provided by means of the PLAXIS procedure for initial stresses. . i. but the HSmodel is not suitable when considering creep. Assuming the secondary compression (for instance during a period of 10 or 30 years) to be a percentage of the primary compression. The special features of these materials is their high degree of compressibility. it is clear that creep is important for problems involving large primary compression. the above stiffness law reduces to: ∗ Eoed = σ / λ where λ∗ = p ref / E oed ref ! = λ* σ For this special case of m = 1. in oedometer testing normally consolidated clays behave ten times softer than normally consolidated sands. depending on the particular type of clay considered. the modified compression index λ* will be known and the PLAXIS user can compute the oedometer modulus from the relationship: ∗ ref ref E oed = p / λ From the above considerations it would seem that the HS-model is perfectly suitable for soft soils. at least for non-cemented laboratory samples. Indeed. Indeed. the Hardening-Soil model yields ε ! / σ . Indeed. secondary compression. on using an exponent equal to one. For many practical soft-soil studies. This is for instance the case when constructing embankments on soft soils.6 SOFT-SOIL-CREEP MODEL (TIME DEPENDENT BEHAVIOUR) 6. This is best demonstrated by oedometer test data as reported for instance by Janbu in his Rankine lecture (1985).1 INTRODUCTION As soft soils we consider near-normally consolidated clays.e. he reports for normally consolidated clays Eoed = 1 to 4 MPa. Another feature of the soft soils is the linear stress-dependency of soil stiffness. This illustrates the extreme compressibility of soft soils. All soils exhibit some creep. and a linear relationship is obtained for m = 1. The differences between these values and stiffnesses for NC-sands are considerable as here we have values in the range of 10 to 50 MPa. which can be integrated to obtain the well-known logarithmic compression law ε = λ* lnσ for primary oedometer loading. clayey silts and peat. and primary compression is thus always followed by a certain amount of secondary compression. Hence. Considering tangent stiffness moduli at a reference oedometer pressure of 100 kPa. most soft soil problems can be analysed using this model. large . According to the Hardening-Soil model we have: ref Eoed = E oed σ / p ref ( ) m at least for c = 0. but conflicts exist. This is a treacherous situation as considerable secondary compression is not preceded by the warning sign of large primary compression. More mathematical lines of research on creep were followed by. (1985). From this 1D differential equation an extension was made to a 3D-model. This Chapter gives a full description of the formulation of the Soft-Soil-Creep model. computations with a creep model are desirable. one is struck by the fact that it concentrates on behaviour related to step loading. In such cases it is desirable to estimate the creep from FEMcomputations. Bjerrum (1967).settlements in later years. Adachi and Oka (1982) and Borja et al. even though natural loading processes tend to be continuous or transient in nature. Mesri (1977) and Leroueil (1977). 3D-creep should be a straight forward extension of 1D-creep. In addition. attention is focused on the model parameters. but this is hampered by the fact that present 1D-models have not been formulated as differential equations. For more applications of the model the reader is referred to Vermeer et al. Again. a state of normal consolidation may be reached and significant creep may follow. for example. Buisman (1936) was probably the first to propose a creep law for clay after observing that soft-soil settlements could not be fully explained by classical consolidation theory. attention is focused on constant strain rate triaxial tests and undrained triaxial creep tests. Then. Garlanger (1972). Finally. a validation of the 3D model is presented by considering both model predictions and data from triaxial tests. Sekiguchi (1977). This mathematical 3D-creep modelling was influenced by the more experimental line of 1D-creep modelling.2 Stress-dependent stiffness (logarithmic compression behaviour) Distinction between primary loading and unloading-reloading Secondary (time-dependent) compression Memory of pre-consolidation stress Failure behaviour according to the Mohr-Coulomb criterion BASICS OF ONE-DIMENSIONAL CREEP When reviewing previous literature on secondary compression in oedometer tests. as a consequence of the loading. Buisman (1936) . This work on 1D-secondary compression was continued by other researchers including. (1998) and Neher & Vermeer (1998). Here. Some basic characteristics of the Soft-Soil-Creep model are: • • • • • 6. Dams or buildings may also be founded on initially overconsolidated soil layers that yield relatively small primary settlements. For the presentation of the Soft-Soil-Creep model we will first complete the line of 1Dmodelling by conversion to a differential form. for example. 1) where εc is the strain up to the end of consolidation. Based on the work by Bjerrum on creep. Indeed. it is convenient to rewrite this equation as: tc + t ′ ε = ε c − C B log for tc t' > 0 (6. Garlanger (1972) proposed a creep equation of the form: τ c + t′ e = ec − Cα log τc with: Cα = C B (1 + e0 ) for: t' > 0 (6. t the time measured from the beginning of loading. Then all excess pore pressures are zero and one observes pure creep for the other 23 hours of the day.3 are entirely identical when choosing τc = tc. The engineering strain ε is replaced by void ratio e and the consolidation time tc is replaced by a parameter τc.3) It would thus seem that there is no creep in this standard oedometer test. Eqs.tc being the effective creep time. Instead. but this suggestion is entirely false.2 and 6.was probably the first to consider such a classical creep test. For the case that τc ≠ tc differences between both formulations will vanish when the effective creep time t’ increases. Hence one obtains t' = 0 and the log-term drops out of Eq. He proposed the following equation to describe creep behaviour under constant effective stress: t ε = ε c − CB log t c for: t > tc (6. Another slightly different possibility to describe secondary compression is the form adopted by Butterfield (1979): . it follows that such tests have no effective creep time. Therefore we will not make any assumptions about the precise values of τc and tc. 6. For practical consulting. oedometer tests are usually interpreted by assuming tc = 24h. Due to the special assumption that this loading period coincides to the consolidation time tc. as published for instance in 1967. compressive stresses and strains are taken to be negative. tc the time to the end of primary consolidation and CB is a material constant.2) with t' = t . Please note that we do not follow the soil mechanics convention that compression is considered positive. (6. Even highly impermeable oedometer samples need less than one hour for primary consolidation. For further consideration.3) Differences between Garlanger’s and Buisman’s forms are modest. the standard oedometer test is a Multiple Stage Loading Test with loading periods of precisely one day. being indicated in Figure 6. τc is the intercept with the (non-logarithmic) time axis of the straight creep line. the logarithmic small strain supersedes the traditional engineering strain. because both τc and C follow directly when fitting a straight line through the data. The subscript ‘H’ is used for denoting logarithmic strain.6) because then logarithmic strain is approximately equal to the engineering strain.5) with the subscript ‘0’ denoting the initial values.7) which allows one to make use of the construction developed by Janbu (1969) for evaluating the parameters C and τc from experimental data. . as the logarithmic strain measure was originally used by Hencky.4) V 1+ e ε H = ln = ln 1+ e0 V 0 (6. In Janbu’s representation of Figure 6. For small strains it is possible to show that: C= Cα C = B ( 1+ e0 )⋅ ln 10 ln 10 (6. In order to do so we depart from Eq. Both the traditional way. We use this particular symbol. Both Butterfield (1979) and Den Haan (1994) showed that for cases involving large strain. The deviation from a linear relation for t < tc is due to consolidation.1b. τ c + t′ ε H = ε cH − C ln τc where εH is the logarithmic strain defined as: (6. one finds: != −ε C τ c + t′ or inversely: − 1 τ c + t′ = ! ε C (6. ON THE VARIABLES τ c AND ε c 6.3 In this section attention will first be focussed on the variable τc.1b can be used to determine the parameter C from an oedometer test with constant load. as well as the Janbu method of Figure 6. (6.1a.4) By differentiating this equation with respect to time and dropping the superscript ‘H’ to simplify notation. The use of the Janbu method is attractive. Here a procedure is to be described for an experimental determination of this variable. The values σp0 and σpc representing the pre-consolidation pressure corresponding to beforeloading and end-of-consolidation states respectively.10) are depicted in a ε−lnσ diagram. 6.tc -ε τc (a) t’ (b) t Figure 6. one adopts the void ratio e instead of ε. In Figure 6.4 and 6. by an equation of the form: σ′ c εc=εe c + ε c = − A ln − B ln σ 0′ σ pc σ p0 (6.8 it follows that: σ pc σ′ τ c + t′ ε = ε e + ε c = − A ln σ − B ln σ − C ln τc 0′ p0 (6. -1/ε εc 1 C tc C 1 t’ = t . and log instead of ln. and the compression index Cc instead of B.8) Note that ε is a logarithmic strain. (6. In most literature on oedometer testing. The above constants A and B relate to Cr and Cc as: A= Cr ( 1 + e0 )⋅ ln 10 B= − Cr ) (1 + e0 )⋅ ln 10 (C c (6.1 Consolidation and creep behaviour in standard oedometer test Considering the classical literature it is possible to describe the end-of-consolidation strain εc.tc ln t . .9) Combining Eqs.10) where ε is the total logarithmic strain due to an increase in effective stress from σ0' to σ' and a time period of tc+t'.2 the terms of Eq. rather than a classical small strain although we conveniently omit the subscript ‘H’. In the above equation σ0' represents the initial effective pressure before loading and σ' is the final effective loading pressure. and the swelling index Cr instead of A. as it should be recalled that restrictions have been made to creep under constant load. as used in traditional elastoplastic modelling. In addition to this basic concept. In fact. apart from Janbu’s method of experimental determination.10) is classical without adding new knowledge. equation (6. Indeed. For generalising the model.4 DIFFERENTIAL LAW FOR 1D-CREEP The previous equations emphasize the relation between accumulated creep and time. we do not assume an instantaneous plastic strain component. For solving transient or continuous loading problems. the more general problem of creep under transient loading conditions has not yet been addressed. we adopt the basic idea that all inelastic strains are time dependent. (6. such a general equation may not contain t' and neither τc as the consolidation time is not clearly defined for transient loading conditions. 6. we have not been able to find precise information on τc in the literature. No doubt. In a first step we will derive an equation for τc. it is necessary to formulate a constitutive law in differential form. for a given constant effective stress. despite the use of logarithmic strain and ln instead of log. one arrives precisely on the NC-line Up to this point.σ’0 A 1 σp 0 σpc σ’ ln(-σ’) εe c A+B 1 NC-line εc c C ln(1+t’/τc) -ε Figure 6. In addition to Eq. Moreover.10) we therefore introduce the expression: . we adopt Bjerrum’s idea that the pre-consolidation stress depends entirely on the amount of creep strain being accumulated in the course of time. as will be described in this section. In order to find an analytical expression for the quantity τc. For t' + tc = 1 day.2 Idealised stress-strain curve from oedometer test with division of strain increments into an elastic and a creep component. the question on the physical meaning of τc is still open. For non-failure situations as met in oedometer loading conditions. a differential form of the creep model is needed. Hence total strain is the sum of an elastic part εe and a time-dependent creep part εc. 11) − εc σ p = σ p 0 exp B Please note that εc is negative. at the end of the day the sample is again in a state of normal consolidation. . it should be realised that usual oedometer samples consolidate for relatively short periods of less than one hour. This quantity can thus be disregarded with respect to τ and it follows that: C τ σ′ = τc σ pc B or: σ pc C τ c =τ σ′ B (6. Hereafter σp increases further from σpc up to σ' during a relatively long creep period. (6.11 to obtain: σp ′ = − C ln τ c + t εc − εc c = − B ln τc σ pc (6. 6. The time-dependency of the pre-consolidation pressure σp is now found by combining Eqs.14) Hence τc depends both on the effective stress σ' and the end-of-consolidation preconsolidation stress σpc. On entering σp = σ' and t' = τ -tc into Eq. In order to verify the assumption (τc . (6. we have OCR=1 both in the beginning and at the end of the load step. During such a load step σp increases from σp0 up to σpc during the short period of (primary) consolidation.13) It is now assumed that (τc − tc) << τ. so that σp exceeds σp0. Hence. at least when adding knowledge from standard oedometer loading.12) it is found that: σ′ = C ln τ c + τ − t c B ln σ pc τc for: OCR =1 (6. but directly after the short consolidation period the sample is under-consolidated with σp<σ'.tc) << τ. where τ is precisely one day. Considering load steps on the normal consolidation line. In this way of stepwise loading the so-called normal consolidation line (NC-line) with σp = σ' is obtained. In conventional oedometer testing the load is stepwise increased and each load step is maintained for a constant period of tc+t' = τ .12) This equation can now be used for a better understanding of τc.14).10 and 6. we thus find very small τc-values from Eq.σp σ′ ε = ε e + ε c = − A ln − ln B σ p0 σ 0′ → → (6. For the usually very high ratios of B/C ≥ 15. The longer a soil sample is left to creep the larger σp grows. Having derived the simple expression 6. (6. (6.10) is differentiated to obtain: !=ε !e + ε !c = − A ε !′ σ C − σ ′ τ c + t′ (6. it is now possible to formulate the differential creep equation.17) THREE-DIMENSIONAL-MODEL On extending the 1D-model to general states of stress and strain. (6.14) can now be introduced to eliminate τc and σpc and to obtain: ! ′ C σ ′ C σ !=ε !e + ε !c = − A − ε σ′ τ σ p 6. To this end Eq. . Eq.14 for τc.18) In Figure 6. being considered negative in this manual.12) to obtain: ! ′ C σ C σ !=ε !e + ε ! c = − A − pc ε σ′ τc σp with: B −ε σ p = σ p 0 exp B c (6. It thus follows that the assumption (τc− tc) << τ is certainly correct. In fact we have the ellipses from the Modified Cam-Clay-Model as introduced by Roscoe and Burland (1968). These invariants are used to define a new stress measure named peq: eq p = p′ + q 2 M p′ 2 (6.16) Again it is recalled that εc is a compressive strain.3 it is shown that the stress measure peq is constant on ellipses in p-q-plane.15) where τc+ t' can be eliminated by means of Eq.Hence not only tc but also τc tend to be small with respect to τ.5 B (6. the well-known stress invariants for pressure p and deviatoric stress q are adopted. 18) that: NC NC 2 1 + 2 K 0 3 (1 − K 0 ) eq + 2 p = σ ′ NC ) 3 1+ 2 K0 M ( 2 NC 1 + 2 K 0 3( 1 − NC ) eq + 2 K 0 NC pp = σ p 3 M (1 + 2 K 0 ) . attention is now focused on normally consolidated states of stress and strain as met in oedometer NC testing. and ppeq can thus be computed from σp. the equivalent pressure peq is constant along ellipsoids in principal stress space. (6.σ3 M 1 p eq pp eq isotropic stress p = -(σ1 + σ2+ σ3)/3 Figure 6.3. We use the general 3D-definition 2. and ppeq is a generalised pre-consolidation pressure.20) NC σ1'.deviatoric stress q = σ1 .7b) for q. also referred to as critical-state friction angle. On using Eq.19) where ϕcv is the critical-void friction angle. For known values of K 0 . To extend the 1D-theory to a general 3D-theory.7b for the deviatoric stress q and: M= 6 sin ϕ cv 3 − sin ϕ cv (6. In such situations it yields σ2' = σ3' = K 0 σ1' . (6. being simply where σ' = K 0 NC proportional to the one-dimensional one. Omitting the elastic strain in . peq can thus be computed from σ' . and it follows from Eq.3 Diagram of peq-ellipse in a p-q-plane The soil parameter M represents the slope of the so-called ‘critical state line’ as also indicated in Figure 6. (2. we will now change to the material parameters κ*. who fit into the framework of critical-state soil mechanics. µ* = C (1 + νur ) (6. as we have only considered a volumetric creep strain ενc. For formulating such a flow rule. It should be noted that the subscript ‘0’ is once again used in the equations to denote initial conditions and that ενc = 0 for time t = 0. we adopt the view that creep strain is simply a time-dependent plastic strain.κ * . as usually done in plasticity theory. introducing the above expressions for peq and ppeq and writing εν instead of ε it is found that: eq C C p c -ε !v = τ peq p B where: −εv eq eq p p = p p 0 exp B c (6. whilst soft soils also exhibit deviatoric creep strains.17). For introducing general creep strains. so that one has a true extension of the 1D-creep model. it is convenient to adopt the vector notation and considering principal directions: σ = (σ 1 σ 2 σ 3 ) T and: ε = (ε1 ε 2 ε 3 )T where T is used to denote a transpose. Similar to the 1D-model we have both elastic and creep strains in the 3D-model. and a flow rule for the creep part. B and C of the 1D-model. It is thus logic to assume a flow rule for the rate of creep strain.21) changes to become: eq µ p −ε !c v= eq τ pp * λ* − κ * µ* with: −εc eq eq v p p = p p 0 exp *− * λ κ (6. Instead of the parameters A.24) where the elasticity matrix and the plastic potential function are defined as: . this equation reduces to Eq. (6.17). λ* and µ*.22) On using these new parameters. one obtains: != ε !′ + λ !e + ε !c = D σ ε 1 ∂ gc ∂σ ′ (6.23) As yet the 3D-creep model is incomplete. B = λ* .the 1D-equation (6.νur ) A . Conversion between constants follows the rules: * κ = 3 (1 . Eq.21) For one-dimensional oedometer conditions. Using Hooke’s law for the elastic part. (6. Hence similar to Eur. To get a proper 3D-model for the elastic strains as well. it has been shown that the 1D-model can be extended to obtain the 3D-model.6 FORMULATION OF ELASTIC 3D-STRAINS Considering creep strains. Eur is not a new input parameter. 1 -ν ur -ν ur 1 -1 D = 1 -ν ur and: -ν ur E ur 1 ν ν ur ur c eq g =p Hence we use the equivalent pressure peq as a plastic potential function for deriving the individual creep strain-rate components.24 this leads to: c !v ε ∂ peq 1 μ* -1 p !′ − = D σ α ∂σ ′ α τ peq p λ* .25) != D σ !′ + ε where: eq pp -1 ∂ peq ∂σ ′ (6. Together with Eqs. Now it can be derived for the volumetric elastic strain that: .23 and 6. Now it follows from the above equations that: ∂ p eq ∂ p eq ∂ p eq ∂ p eq c c c ! ! ! = = = ε ε ε = = = λ =α + + λ + + λ !c ε v 1 2 3 ∂σ ∂p ′ ∂ σ 2′ ∂σ 3′ 1′ Hence we define α = ∂peq/∂p’. the bulk modulus Kur is stress dependent according to the rule Kur= p' / κ*. but as yet this has not been done for the elastic strains. the elastic modulus Eur has to been defined as a stress-dependent tangent stiffness according to: E ur = 3(1 − 2ν ur )K ur = 3(1 − 2ν ur ) p′ κ * (6. but simply a variable quantity that relates to the input parameter κ*. On the other hand νur is an additional true material constant. The subscripts ‘ur’ are introduced to emphasize that both the elasticity modulus and Poisson’s ratio will determine unloading-reloading behaviour.26) = eq p p0 − εc v exp *− * λ κ or inversely: − c εv = (λ − κ * * ) peq p ln eq p p0 6. 6.27) Hence.κ* eq μ* (6. However mean stress can be converted into principal stress. and the dilatancy angle ψ. the Soft-SoilCreep model requires the following material constants: Failure parameters as in the Mohr-Coulomb model: c : Cohesion : Friction angle : Dilatancy angle [kN/m2] [°] [°] ϕ ψ . we NC NC have both -3p'= (1+2 K 0 )σ' and -3p0' = (1+2 K 0 )σ0' and it follows that p'/p0 = σ'/σ0. As a consequence we derive the simple rule -ενc =κ* ln(σ'/σ0'). rather than by principal stress σ' as in the 1D-model. instantaneous plastic strain ! p = λ ∂g/∂σ' with g = g (σ'.ϕ) = 0 is met.!e ε v= !′ p K ur = − κ* !′ p p′ or by integration: p′ * -εe v = κ ln p0′ (6.30) where K0 depends to a great extent on the degree of overconsolidation. the dilatancy angle tends to be small. ψ). It would thus seem that κ* coincides with A. This gives as rates develop according to the flow rule ε additional soil parameters the effective cohesion. For fine grained.29) !′ 1 + ν ur p = 1 − ν ur p′ κ 1 + 2K0 * !′ σ σ′ (6. In conclusion. c. cohesive soils. ϕ.28) Hence in the 3D-model the elastic strain is controlled by the mean stress p'. c. Unfortunately this line of thinking cannot be extended towards overconsolidated states of stress and strain. it can be derived that: !′ 1 + ν ur p = 1 − ν ur p′ and it follows that: * . For one-dimensional compression on the normal consolidation line. whereas the 1D-model involves -ενc =A ln(σ'/σ0'). Good agreement with the 1D-model is thus found by taking κ*≈2A. For such situations.ε !e v = κ 1 1 + 2K0 !′ σ σ′ (6. the Mohr-Coulomb friction angle. For many situations. it is reasonable to assume K0 ≈ 1 and together with νur ≈ 0.2 one obtains -2ενc ≈ κ* ln(σ'/σ0'). 6. it may often be assumed that ψ is equal to zero.7 REVIEW OF MODEL PARAMETERS As soon as the failure yield criterion f (σ'. 19).1°. In addition.4 Parameters tab for the Soft-Soil-Creep model By default.Basic stiffness parameters: κ* λ* µ * : Modified swelling index : Modified compression index : Modified creep index [-] [-] [-] Advanced parameters (it is advised to use the default setting): νur NC K0 : Poisson's ratio for unloading-reloading (default 0. approximate value of K 0 NC resulting default values of K 0 tend to be somewhat higher than the ones that follow NC NC may be entered after from Jaky's formula K 0 = 1–sinϕ. (6. M is calculated from Eq.15) : σ'xx / σ'yy stress ratio in a state of normal consolidation NC -related parameter (see below) : K0 [-] [-] [-] M Figure 6. 1994): . Alternatively. In general. values of K 0 which the corresponding value of M is calculated from the relation (Brinkgreve. using ϕcν = ϕ + 0. this is not an experimental finding. PLAXIS displays the NC that corresponds to the default setting of M. but just a practical default value. 1). Instead he can choose NC . The parameter µ* can be obtained by measuring the volumetric strain on the long term and plotting it against the logarithm of time (see Figure 6.6. modified compression index and modified creep index These parameters can be obtained both from an isotropic compression test and an oedometer test. and the slope of the unloading (or swelling) line can be used to compute the modified swelling index κ*.2). Note that there is a difference between the modified indices κ* and λ* and the original Cam-Clay parameters κ and λ. as explained in section 6. . the plot can be approximated by two straight lines (see Figure 6. When plotting the logarithm of stress as a function of strain. The slope of the normal consolidation line gives the modified compression index λ*.5 Advanced parameters for Soft-Soil-Creep model Modified swelling index. values for K 0 Figure 6. The latter parameters are defined in terms of the void ratio e instead of the volumetric strain εν.31) Hence the user can not enter directly a particular value of M.M =3 (1 − K ) + (1 − K )(1 − 2ν )(λ / κ − 1) (1 + 2K ) (1 + 2K )(1 − 2ν ) λ / κ − (1 − K )(1 + ν NC 2 0 NC 2 0 NC 0 NC 0 ur * ur * * * NC 0 ur ) (6. For the approximation it is assumed that the average stress state during unloading is an isotropic stress state. Poisson's ratio In the case of the Soft-Soil-Creep model. Poisson's ratio plays a minor role. be computed from a given value of the overconsolidation ratio (OCR).6. but it becomes important in unloading problems.Table 6. the horizontal and vertical stresses are equal. there is no exact relation between the isotropic compression indices κ and κ* and the one-dimensional swelling indices Ap and Cr.3(1 + e ) κ * ≈ 2 Cr 2. Subsequently σp0 can be used to compute the initial value of the generalised pre-consolidation pressure ppeq (see Section 2. it is also necessary to know the initial pre-consolidation pressure σp0. the fact that λ* / µ* is in the range between 15 to 25 and the general observation λ*/κ* is in the range between 5 to 10.15 is automatically adopted. As . i. For example. This pressure may. For a rough estimate of the model parameters.3 1 + e * μ = Cα 2.2.3(1 + e ) In Table 6. As already indicated in section 6. because the ratio of horizontal and vertical stress changes during one-dimensional unloading. one might use the correlation λ*≈Ip(%)/500. Ap represents the slope of the one-dimensional unloading curve in the ln p'-εν plot. for example.1a Relationship to Cam-Clay parameters * λ = λ 1+e * κ = κ 1+e --- Table 6. Its value will usually be in the range between 0.1c Relationship to internationally normalized parameters * λ = Cc 2.1 and 0. the relatively small Poisson's ratio will result in a small decrease of the lateral stress compared with the decrease in vertical stress. For loading of normally consolidated materials. then the value νur = 0. If the standard setting for the Soft-SoilCreep model parameters is selected. For characterising a particular layer of soft soil.1b. Poisson's ratio is purely an elasticity constant rather than a pseudo-elasticity constant as used in the Mohr-Coulomb model.6).3 C s’ Table 6.e.1b Relationship to Dutch engineering practice * λ = 1 C p’ κ * ≈ 2 Ap * μ ≈ 1 2. for unloading in a onedimensional compression test (oedometer). 04 0. the ratio of horizontal and vertical stress increases.32) VALIDATION OF THE 3D-MODEL This section briefly compares the simulated response of undrained triaxial creep behaviour of Haney clay with test data provided by Vaid and Campanella (1977).10 % per minute 0 0. but on the ratio of difference in horizontal stress to difference in vertical stress in oedometer unloading and reloading: ∆σ xx νur = ∆σ yy 1 − νur 6. which is a well-known phenomenon for overconsolidated materials. All triaxial tests considered were completed by initially consolidating the samples under an effective isotropic confining pressure of 525 kPa for 36 hours and then allowing them to stand for 12 hours under undrained conditions before starting a shearing part of the test.08 0.15 % per minute 1.8 (unloading and reloading) (6. deviatoric stress q=(σ1-σ3) 400 300 200 Soft Soil Creep model 100 0.a result. using the material parameters summarized below.6 Results of the undrained triaxial tests (CU-tests) with different rates of strain The faster the test the higher the undrained shear strength .00094 % per minute 0. Hence.12 axial strain ε1 Figure 6. An extensive validation of the Soft-SoilCreep model is also provided in Stolle et al.00 0. Poisson's ratio should not be based NC on the normally consolidated K 0 -value. (1997). 25 The end-of-consolidation pre-consolidation pressure.1o λ* = 0.105 ψ = 0o NC K 0 = default µ* = 0. It is observed that the model describes the tests very well. The slower a test is performed. deviatoric stress q = σ1 . This value was determined by simulating the consolidation part of the test. Constant strain rate shear tests Undrained triaxial compression tests. It is clear that the pre-consolidation pressure not only depends on the applied maximum consolidation stress.7 Strain rate dependency of the effective stress path in undrained triaxial tests !v = ε !v e + ε !v c = 0 or equivalently ε !v e = −ε !v c . In Figure 6. This behaviour is shown in Figure 6. which would have been required for an OCRo of 1.σ3 “Fast” tests “Slow” tests isotropic stress p = -(σ1 + 2σ3)/3 Figure 6. but also on creep time as discussed in previous sections.The material properties for Haney Clay are: κ*= 0. The . the larger the creep compaction is and finally the elastic swelling. creep compaction is balanced by elastic swelling of the sample.7. The preconsolidation pressure ppeq of 373 kPa is less than 525 kPa. performed under !1 and constant horizontal pressure σ3 where the shear constant rates of vertical strain ε stress q can move. was found to be 373 kPa.016 φmc = 32o φcs = 32.004 c = 0 kPa ν = 0.6 we can see the results of Vaid and Campanella’s tests (1977) for different strain rates and the computed curves. ppeq.6. that were calculated with the present creep model. During these undrained tests it yields ε Hence. as considered in Figure 6. Undrained triaxial creep tests These tests begin with isotropic consolidation up to a mean stress of 525 kPa. !v c = 0 and consequently For extremely fast tests there is no time for creep.4 q = 300. Finally all external stresses are kept constant and the sample is subjected to undrained creep.3 q = 278. This agrees well with the experimental data summarized by Kulhawy and Mayne (1990).04 0. Hence in this extreme case there is no elastic volume change and also ε consequently neither a change of the mean stress. On inspecting all numerical results. it appears that the undrained shear strength. This implies a straight vertical path for the effective stress in p-q-plane.00 1 10 100 1000 10000 time [minutes] .06 q = 323. Then a deviatoric stress q is applied in undrained loading. where K ur is the elastic bulk modulus.02 + 0. may be approximated by the equation: cu ≈ 1. shows that elastic ! ' = K ur ε expression p swelling implies a decrease of the mean stress.!v e . axial strain ε1 0.09 log ε ! * cu (6.33) where cu* is the undrained shear strength in an undrained triaxial test with a strain rate of 1% per hour.08 0.3 0.02 experimental Soft Soil Creep model 0. it yields ε !v e = 0 . cu. Then undrained samples were loaded up to different deviatoric stresses.σ3) [kPa] Figure 6.e. Creep under constant deviatoric stress is observed. being well predicted by the Soft-Soil-Creep model. Figure 6. For relatively small stress ratios. Creep rupture time [minutes] 10000 experimental Soft Soil Creep model 1000 100 10 1 260 280 300 320 340 deviatoric stress q=(σ1. as indicated by the asymptotes in Figure 6. The amount of creep depends on the applied deviatoric stress q or rather on the applied stress ratio q/p.8 shows the actual creep strain evolution for samples with three different deviatoric stresses. For large stress ratios. The creep rupture time is the creep ! = ∞ .9 Results of triaxial creep tests.Samples were first consolidated under the same isotropic stress. All tests have different constant deviatoric stress.8. creep rates are small and they decrease in course of time. creep rates increase with time and samples will finally fail. however. strain rates become infinitely large. i. time up to a creep rate ε . . Distinction between primary loading and unloading-reloading. it was decided to keep the Soft-Soil model in PLAXIS Version 8. This has resulted into the current Hardening-Soil model. (7. The difference is that Eq.1) a minimum value of p' is set equal to a unit stress. The parameter λ* is the modified compression index. Memory for pre-consolidation stress. which determines the compressibility of the material in primary loading. p'.1) gives a straight line as shown in Figure 7.1) In order to maintain the validity of Expression (7. which can be formulated as: p’ * εv . ISOTROPIC STATES OF STRESS AND STRAIN (σ 1' = σ 2' = σ 3') In the Soft-Soil model. the idea of using separate models for soft soil and hard soil has been excluded.1 Stress dependent stiffness (logarithmic compression behaviour). εν. Instead. Note that λ* differs from the index λ as used by Burland (1965).ε0 v = . Some features of the Soft-Soil model are: • • • • 7. Failure behaviour according to the Mohr-Coulomb criterion. Plotting Eq. the Soft-Soil model can be substituted by the new Hardening-Soil model or the Soft-Soil-Creep model. As a result. it is assumed that there is a logarithmic relation between the volumetric strain. .1. and the mean effective stress. Up to Version 6 PLAXIS material models had consisted of Mohr-Coulomb model. in order not to deny users’ preferences to use models that they have got to know well. however. it is worth mentioning that starting from Version 7 some changes to the soil modelling strategy of PLAXIS have been introduced. At the same time the Soft-Soil-Creep model was implemented to capture some of the very special features of soft soil. In Version 7.7 THE SOFT SOIL MODEL To highlight the significance of the Soft-Soil model. However. (7.λ ln 0 p (virgin compression) (7. Soft-Soil model and Hard-Soil model. the Hard-Soil model was further developed to become an advanced model for soils ranging from soft to hard.1) is a function of volumetric strain instead of void ratio. 2) Again. however. each corresponding to a particular value of the isotropic pre-consolidation stress pp. a minimum value of p' is set equal to a unit stress. The ratio λ*/κ* is. Note that effective parameters are considered rather than undrained soil properties.3) in which the subscript ur denotes unloading / reloading. which determines the compressibility of the material in unloading and subsequent reloading. equal to Burland's ratio λ /κ.κ ln p0 (unloading and reloading) (7.1 Logarithmic relation between volumetric strain and mean stress During isotropic unloading and reloading a different path (line) is followed. The pre- . Eur. Note that κ* differs from the index κ as used by Burland. (7. is used as an input parameter. The parameter κ* is the modified swelling index. as might be suggested by the subscripts ur. which can be formulated as: p’ e0 * εe v .2) implies linear stress dependency on the tangent bulk modulus such that: K ur ≡ p’ Eur = * 3 (1 .ε v = .1. The soil response during unloading and reloading is assumed to be elastic as denoted by the superscript e in Eq.2). νur and κ* are used as input constants for the part of the model that computes the elastic strains.2 ν ur ) κ (7. An infinite number of unloading / reloading lines may exist in Figure 7.2) and Eq. (7. Instead. Kur. Neither the elastic bulk modulus. nor the elastic Young's modulus. The elastic behaviour is described by Hooke's law (see Section 2.Figure 7. 1967) the M-line is referred to as the critical state line and represents stress states at post peak failure. restriction is made to triaxial loading conditions under which σ2' = σ3'.pp (7.consolidation stress represents the largest stress level experienced by the soil. which might not correspond to the M-line. failure is not necessarily related to critical state. the ellipse extends to c cotϕ . for clarity. The parameter M is then based on the critical state friction angle. In view of this. In the Soft-Soil model. In tension (p' < 0). As a result. In primary loading.2. YIELD FUNCTION FOR TRIAXIAL STRESS STATE (σ 2' = σ 3') 7.ε vp exp p p = p0 p * . the value of M can be chosen such that a known value of K0nc is matched in primary one-dimensional compression.q) and pp. During loading. For such a state of stress the yield function of the Soft-Soil model is defined as: f = f . the pre-consolidation stress. as illustrated in Figure 7. infinitely many ellipses may exist (see Figure 7. However.5) determines the height of the ellipse. in this section. In the Modified Cam-Clay model (Burland 1965. but it ensures a proper matching of K0nc. however. K0nc.* λ κ (7. this pre-consolidation stress remains constant.6) The yield function f describes an ellipse in the p'-q-plane. is a function of plastic strain such that: f = q + p’ 2 M (p’ + c cot ϕ ) 2 (7. Such an interpretation and use of M differs from the original critical state line idea. causing irreversible (plastic) volumetric strains. however. pp. The tops of all ellipses are located on a line with slope M in the p'-q-plane.2 The Soft-Soil model is capable to simulate soil behaviour under general states of stress. determines the extent of the ellipse along p' axis. (7. the parameter M determines largely the coefficient of lateral earth pressure. The isotropic pre-consolidation stress. The parameter M in Eq.4) where f is a function of the stress state (p'. the pre-consolidation stress increases with the stress level. The Mohr-Coulomb failure criterion is a function of the strength parameters ϕ and c.5) . The height of the ellipse is responsible for the ratio of horizontal to vertical stresses in primary onedimensional compression.2) each corresponds to a particular value of pp. During unloading and reloading. the plastic behaviour of the Soft-Soil model is defined by a total of six yield functions. In the Soft-Soil model. a perfectly-plastic Mohr-Coulomb type yield function is used. three compression yield functions and three Mohr-Coulomb yield functions.5) and Figure 7.2).2. The slope of the failure line is smaller than the slope of the M-line.6) the initial volumetric plastic strain is assumed to be zero.6. The total yield contour. the 'cap') will remain in the 'compression' zone (p' > 0) a minimum value of c cotϕ is adopted for pp. Eq. (7. and forms the cap of the yield contour. (7. Eq. is the boundary of the elastic stress area. In order to make sure that the right hand side of the ellipse (i.3. there is a 'threshold' ellipse as illustrated in Figure 7. The total yield contour in principal stress space. The failure line is fixed. as shown by the bold lines in Figure 7. pp0 can be regarded as the initial value of the pre-consolidation stress. resulting from these six yield functions. For c = 0. .(Eq. describes the irreversible volumetric strain in primary compression. To model the failure state. but the cap may increase in primary compression. Stress paths within this boundary give only elastic strain increments. (7. Hence. This yield function represents a straight line in p'-q-plane as shown in Figure 7.2.e. a minimum value of pp equal to a stress unit is adopted.2 Yield surface of the Soft-Soil model in p'-q-plane The value of pp is determined by volumetric plastic strain following the hardening relation. whereas stress paths that tend to cross the boundary generally give both elastic and plastic strain increments. According to Eq.2. The determination of pp0 is treated in Section 2. is indicated in Figure 7. the yield function. For general states of stress. This equation reflects the principle that the pre-consolidation stress increases exponentially with decreasing volumetric plastic strain (compaction). (7.4).6). Figure 7. the modified creep index µ* is not considered. since the Soft-Soil model does not include time. the Soft-Soil model requires the following material constants: Basic parameters: λ* κ c * : : : : : : : Modified compression index Modified swelling index Cohesion Friction angle Dilatancy angle Poisson's ratio for unloading / reloading Coefficient of lateral stress in normal consolidation NC [-] [-] [kN/m2] [°] [°] [-] [-] ϕ ψ νur K0NC Advanced parameters (use default settings): .−σ’1 cap failure surface −σ’3 −σ’2 Figure 7. However.3 PARAMETERS OF THE SOFT-SOIL MODEL The parameters of the Soft-Soil model coincide with those of the Soft-Soil-Creep model. Thus.3 Representation of total yield contour of the Soft-Soil model in principal stress space 7. Note that there is a difference between the modified indices κ* and λ* and the original Cam-Clay parameters κ and λ. Another relationship exists with the Dutch parameters for one-dimensional compression. Note that.Figure 7. the plot can be approximated by two straight lines (see Figure 7. The latter parameters are defined in terms of the void ratio e instead of the volumetric strain εν. Cc and Cr. M is calculated automatically from the coefficient of the lateral earth pressure. physically. . the parameters κ* and λ* can be obtained from the one-dimensional compression test. Apart from the isotropic compression test.4 shows PLAXIS window for inputting the values of the model parameters. in the current model M differs from that in the Modified Cam-Clay model where it is related to the material friction. and the slope of the unloading (or swelling) line gives the modified swelling index. (7.4 Parameters tab for the Soft-Soil model Modified swelling index and modified compression index These parameters can be obtained from an isotropic compression test including isotropic unloading. K0NC. Cp' and Ap. by means of Eq. Here a relationship exists with the internationally recognized parameters for one-dimensional compression and recompression. These relationships are summarized in Table 7.1). When plotting the logarithm of the mean stress as a function of the volumetric strain for clay-type materials.8).1. Figure 7. The slope of the primary loading line gives the modified compression index. This means that entering a cohesion larger than zero may result in a state of 'over-consolidation'.1: • In relations 1 and 2.Table 7. In fact. the cap) in the 'pressure' zone of the stress space. Entering a cohesion will result in an elastic region that is partly located in the 'tension' zone.e. a stiffer behaviour is obtained during the onset of loading.1a Relationship to Cam-Clay parameters 1. κ * ≈ 2 Cr 2. * λ = 1 C p’ 4.3 (1 + e ) Remarks on Table 7. e. For approximation it is assumed that the average stress state during unloading is an isotropic stress state. In relations 4 and 6 there is no exact relation between κ* and the one-dimensional swelling indices. but this will give a relatively small difference in void ratio. The left hand side of the ellipse crosses the p'axis at a value of -c cotϕ. the void ratio. As a result. * λ = Cc 2. When using the standard setting the cohesion is set equal to 1 kPa.1c Relationship to internationally normalized parameters 5. is assumed to be constant. depending on the magnitude of the cohesion and the initial stress state. Any effective cohesion may be used. * λ = λ 1+e 2 * κ = κ 1+e Table 7. It is not possible to specify undrained shear strength by means of high cohesion .2.e. The ratio λ*/κ* (=λ /κ) ranges. • • • Cohesion The cohesion has the dimension of stresses. the isotropic pre-consolidation stress pp has a minimum value of c cotϕ. because the ratio of horizontal and vertical stresses changes during one-dimensional unloading. between 3 and 7.3 in relation 5 is obtained from the ratio between the logarithm of base 10 and the natural logarithm. i. as illustrated in Figure 7. including a cohesion of zero. In order to maintain the right hand side of the ellipse (i. the horizontal and vertical stresses are equal. The factor 2. e will change during a compression test. in general. κ * ≈ 2 Ap Table 7.1b Relationship to Dutch engineering practice 3. For e one can use the average void ratio that occurs during the test or just the initial value.3 ( 1 + e) 6. For loading of normally consolidated materials.15 is automatically used. the Poisson's ratio is the well known pure elastic constant rather than the pseudo-elasticity constant as used in the Mohr-Coulomb model. using a high friction angle will substantially increase the computational requirements. Input of model parameters should always be based on effective values. It is often recommended to use ϕcν. Moreover.7) The parameter M is automatically determined based on the coefficient of lateral earth pressure in normally consolidated condition. as entered by the user. Dilatancy angle For the type of materials. i. As a result. but on the ratio of the horizontal stress increment to the vertical stress increment in oedometer unloading and reloading test such that: ν ur = ∆σ xx 1 -ν ur ∆σ yy K0 NC-parameter (unloading and reloading) (7.e. It is specified in degrees. care should be taken with the use of high friction angles. which can be described by the Soft-Soil model. Friction angle The effective angle of internal friction represents the increase of shear strength with effective stress level. Zero friction angle is not allowed. then νur = 0. for unloading in a one-dimensional compression test (oedometer). K0NC. the dilatancy can generally be neglected.1 and 0. If the standard setting for the Soft-Soil model parameters is selected. the critical state friction angle. Its value will usually be in the range between 0. A dilatancy angle of zero degrees is considered in the standard settings of the Soft-Soil model.and a friction angle of zero. Hence. Poisson's ratio In the Soft-Soil model. rather than a higher value based on small strains. Poisson's ratio should not be based on the normally consolidated K0NC-value. the ratio of horizontal and vertical stress increases. the relatively small Poisson's ratio will result in a small decrease of the lateral stress compared with the decrease in vertical stress.2. For example. On the other hand. Poisson's ratio plays a minor role. which is a well-known phenomenon in overconsolidated materials. 1994): . but it becomes important in unloading problems. The exact relation between M and K0NC gives (Brinkgreve. 2.9) .8) The value of M is indicated in the input window.0 .8K 0NC (7.8). M is also influenced by the Poisson's ratio νur and by the ratio λ*/κ*. the influence of K0NC is dominant.8) can be approximated by: M ≈ 3. Eq. However.M ≈3 (1 − K ) + (1 − K )(1 − 2ν )(λ /κ − 1) ( (1 + 2K ) 1 + 2K )(1 − 2ν )λ /κ − (1 − K )(1 + ν NC 2 0 NC 2 0 NC 0 NC 0 ur ∗ ur ∗ ∗ ∗ NC 0 ur ) (7. (7. (7. As can be seen from Eq. . 2 0. but they are selected for simplicity.9 Medium 30000 90000 30000 0.5 0.8 APPLICATIONS OF ADVANCED SOIL MODELS In this chapter. These dimensions are not realistic.0 40 10 0.1 HS MODEL: RESPONSE IN DRAINED AND UNDRAINED TRIAXIAL TESTS In this Section. Table 8. The left hand side and the bottom of the geometry are axes of symmetry. provided that the soil weight is not taken into account. For applications on the standard MohrCoulomb model. 8. Figure 8. In this configuration the stresses and strains are uniformly distributed over the geometry. the reader is referred to the Tutorial Manual.43 0.0 35 5 0.36 0. .5 0.5 0.0 0. The remaining boundaries are fully free to move.9 Unit kN/m2 kN/m2 kN/m2 kN/m2 ° ° kN/m2 - A triaxial test can simply be modelled by means of an axisymmetric geometry of unit dimensions (1m x 1m).1 Arbitrary Hardening-Soil parameters for sands of different densities Parameter E50ref (for pref = 100 kPa) Eurref (for pref = 100 kPa) Eoedref (for pref = 100 kPa) Cohesion c Friction angle ϕ Dilatance angle ψ Poisson's ratio νur Power m K0nc (using Cap) Tensile strength Failure ratio Loose 20000 60000 20000 0.0 30 0 0. The dimension of the model does not influence the results. representing sands of different properties.5 0. Table 8. advanced soil models will be utilised in various applications in order to illustrate the particular features of these models. are considered. The deformation magnitudes in x.1.0 0.1. the Hardening-Soil model is utilised for the simulations of drained and undrained triaxial tests.2 0. At these boundaries the displacements normal to the boundary are fixed and the tangential displacements are kept free to allow for 'smooth' movements.9 Dense 40000 120000 40000 0.0 0.2 0. Arbitrary sets of model parameters.and y-direction of the top right hand corner correspond to the horizontal and vertical strains respectively. that represent a quarter of the soil specimen. Table 8. and as described in the Reference manual. Figure 8. the confinement pressure p' is applied by activating load A and B. This implies modification of load A by double clicking the load in the geometry model. the load configurations and magnitudes can be specified in the Input program. In the second phase the displacements are reset to zero and the sample is vertically loaded up to failure while the horizontal load is kept constant. However. Figure 8. the failure level is higher when the sand is denser. For this case. The HS model does not include softening behaviour.) The latter phase is carried out for drained as well as undrained conditions. which is typical for the Hardening-Soil model. A very course mesh is sufficient for this simple geometry.As in Version 7 the value of the applied loads can be controlled by the load multipliers such as ΣMloadA and ΣMloadB. the calculation of all phases can be done by means of the Staged construction process. These calculations are performed for the three different sets of material parameters. and to simulate the confining pressure p'. The computational results are presented in the figures on the following pages. Then in the calculation program these loads can be activated or deactivated by means of the Staged construction option. Obviously.1. This shows a hyperbolic relationship between the stress and the strain.2 shows the principal stress difference versus the axial strain for the drained condition.1 Simplified configuration of a triaxial test In the Calculation program. at least in the drained tests. as shown in Figure 8. . Initial stresses and steady pore pressures are not taken into account. (Details of the procedure can be found in the Reference and Tutorial manuals.1. in Version 8. As a result a load window appears in which the input values of the load can be changed. so after reaching failure the stress level does not reduce. In the first phase. distributed loads of -100 kN/m2 representing the principal stresses σ1’ (load A) and σ3’ (load B) are applied in the Input program. 04 0. This graph clearly shows the influence of dilatancy in the denser sands.ε1 Figure 8.2 σ1-σ3 [kN/m ] 400 dense 300 medium 200 loose 100 0 0 0.01 0.05 0. .000 -0. the transition from elastic behaviour to failure is much more gradual when using the Hardening-Soil model. Principal stress difference versus axial strain εv 0. In contrast to the Mohr-Coulomb model.010 0 0.3 shows the axial strain versus the volumetric strain for the drained test.06 loose dense .005 medium 0.010 0.04 0.02 0.02 0.005 -0.3 Results of drained triaxial tests using the Hardening-Soil model. plastic strain occurs immediately after load application.06 ε1 Figure 8.2 Results of drained triaxial tests using the Hardening-Soil model. in the HS model.03 0. Volumetric strain versus axial strain Figure 8.03 0.015 0.05 0.01 0. In fact. Principal stress difference versus axial strain pexcess [kPa] -60 loose -50 -40 medium -30 dense -20 -10 0 0 5.σ1-σ3 [kN/m ] 200 2 160 dense 120 medium loose 80 40 0 0 5.4 Results of undrained triaxial tests using the Hardening-Soil model.00E-03 0.020 0.020 0.015 0.010 0.030 -ε1 Figure 8.015 0.010 0.030 -ε1 Figure 8.00E-03 0.025 0. Excess pore pressure vs axial strain .025 0.5 Results of undrained triaxial tests using the Hardening-Soil model. during both the drained and undrained tests. for the medium sand. However. lower than that of the drained tests. both tests were drained. for the medium and dense sands the stress level continues to increase after reaching the failure level due to the fact that dilatancy occurs which causes reduction of excess pore pressures and thus increase of the effective stresses. In the second phase there is a clear distinction between the two tests.2 APPLICATION OF THE HARDENING-SOIL MODEL ON REAL SOIL TESTS In this section the ability of the Hardening-Soil model to simulate laboratory tests on sand is examined by comparing PLAXIS calculation results with those obtained from laboratory tests. in principle. The decrease in horizontal effective stress is more than when if the Mohr-Coulomb model would have been used. Figure 8.6 shows the effective stress paths. This is attributed to the plastic compaction (Cap hardening) that occurs in the HS model. In the undrained test the effective horizontal stress reduces while the vertical stress increases due to the development of excess pore pressures.In the undrained tests. On the basis of these tests the model parameters for the Hardening-Soil model .6 Stress paths for drained and undrained triaxial tests using the HS model Figure 8. Extensive laboratory tests were conducted on loose and dense Hostun sand. Figure 8. 8. During first phase (isotropic loading). This can be seen in Figure 8. the failure level is.5.4. 44 0.0 0. However. It can be seen that the material response (measured and computed) show gradual transition from elastic to plastic behaviour. Modelling of the softening behaviour. in the Hardening-Soil model the dilatancy continues to infinity. unless the dilatancy cut-off option has been used.5 37000 90000 29600 0. and thus. It can also be seen from the test data that the dilatancy reduces during softening.34 0.8 and 8-8.0 41 14 0.0 0. the deviatoric stress remains constant.20 0.7. In the first phase the sample is isotropically compressed up to a confining pressure of p' = -300 kN/m2. Figure 8. In PLAXIS the procedure for the simulation of the triaxial tests has been described in Section 8.1. The failure level is fully controlled by the friction angle (the cohesion is zero). As such the relation between the deviatoric stress and the axial strain can be approximated by a hyperbola.2 Hardening-Soil parameters for loose and dense Hostun sand Parameter Volumetric weight γ E50ref (pref = 100 kPa) Eurref (pref = 100 kPa) Eoedref (pref = 100 kPa) Cohesion c Friction angle ϕ Dilatancy angle ψ Poisson's ratio νur Power m K0nc Tensile strength Failure ratio Loose sand 17 20000 60000 16000 0.0 34 0 0.Table 8.9 Dense sand 17.20 0.9 Unit kN/m3 kN/m2 kN/m2 kN/m2 kN/m2 ° ° kN/m2 - Triaxial test Standard drained triaxial tests were performed on loose and dense sand specimens. The figures show that the computational results match reasonably with the test data. is not incorporated in the HardeningSoil model. .65 0. however. In the second phase the sample is vertically loaded up to failure while the horizontal stress (confining pressure) is kept constant. The computational results and the measured data are presented in Figure 8.50 0. The test results on dense sand show softening behaviour after the peak load has been reached. 7 Results of drained triaxial tests on loose Hostun sand.8 Results of drained triaxial tests on dense Hostun sand. principal stress ratio versus axial strain . principal stress ratio versus axial strain |σ1-σ3| [kPa] 1400 1200 1000 800 600 400 200 0 0 5 10 15 Hardening soil model test data -ε1 [%] Figure 8.|σ1-σ3| [kPa] 1000 800 600 400 200 0 0 5 10 15 15 Hardening soil model test data εv [%] 1 0 -1 -2 0 5 -ε1 [%] 10 Figure 8. a set of oedometer test on both loose and dense sands. A coarse mesh is sufficient for this case.10 Simplified configuration of an oedometer test . volumetric strain versus axial strain Oedometer test As for the triaxial test. In PLAXIS the oedometer test is simulated as an axisymmetric geometry with unit dimensions. was conducted.9 Results of drained triaxial tests on dense Hostun sand. Table 8.10. Figure 8. Figure 8.εv [%] 8 7 6 5 4 3 Hardening soil model 2 1 0 -1 -2 0 5 10 15 test data -ε1 [%] Figure 8.2. 02 0.12.The computational results as compared with those obtained from the laboratory tests are shown in Figure 8.11 and Figure 8.005 0.015 0.11 Results of oedometer test on loose Hostun sand. axial stress versus axial strain -σyy [kPa] 400 Hardening soil model 300 test data 200 100 0 0 0.02 0. -σyy [kPa] 400 300 Hardening soil model test data 200 100 0 0 0.12 Results of oedometer test on dense Hostun sand.01 0.015 0.025 -εyy Figure 8. axial stress versus axial strain .005 0.01 0.025 -εyy Figure 8. From a stress free state the loose sand sample is loaded consecutively to 25 kPa, 50 kPa, 100 kPa and 200 kPa with intermediate unloading. The dense sand sample is loaded to 50 kPa, 100 kPa, 200 kPa and 400 kPa with intermediate unloading. As it can be seen, the computational results show a reasonable agreement with the test data. No doubt, distinction should be made between loose and dense soil, but it seems that for a soil with a certain density the stiffness behaviour under different stress paths can be well captured with a single set of model parameters. (A small offset of 0.15% has been applied to the computational results of the loose sample in order to account for the relative soft response at the beginning of the test.) Pressiometer test In this Section the Pressiometer test is simulated and results from PLAXIS and laboratory experimentation are compared. Laboratory testing results on dense sand with material parameters listed in Table 8.2 are used. In the field, the pressiometer with 44 mm in diameter covered with a membrane with 160 mm in height is attached to the Cone penetration shaft. In the laboratory, the pressiometer is attached to a 44 mm pipe and placed in a circular calibration chamber with a diameter of 1.2 m and a height of 0.75 m. A large overburden pressure of 500 kPa is applied at the surface to simulate the stress state at larger depths. In PLAXIS only half of the geometry is simulated by an axisymmetric model, Figure 8.13. The overburden pressure is simulated by load A, and the expansion of the pressiometer is simulated by imposing a horizontal distributed load, load B. Therefore the initial standard boundary conditions have to be changed near the pressiometer in order to allow for free horizontal displacements. Figure 8.13 Geometry model for pressiometer test To allow for a discontinuity in horizontal displacements, a vertical interface along the shaft of the pressiometer borehole and a horizontal interface just above the pressiometer are introduced. Both interfaces are set rigid (Rinter = 1.0). Extra geometry lines are created around the pressiometer to locally generate a finer mesh. After the generation of initial stresses, the vertical overburden load (load A) is applied using the standard boundary fixities. From the calculations, the lateral stress around the pressiometer appears to be 180 kPa. Subsequently, the horizontal fixity near the pressiometer is removed, in the Input program, and replaced by Load B with a magnitude of 180 kPa. In the next calculation the pressure (load B) is further increased by use of Staged construction in an Updated Mesh analysis. The results of this calculation are presented in Figure 8.14 and Figure 8.15. Figure 8.14 Stress distribution in deformed geometry around the pressiometer at a pressure of 2350 kPa Figure 8.14 shows details of the deformations and the stress distribution when the pressure in the pressiometer was 2350 kPa. The high passive stresses appear very locally near the pressiometer. Just above the pressiometer the vertical stress is very low due to arching effects. Away from the pressiometer, a normal K0-like stress state exists. Figure 8.15 shows a comparison of the numerical results with those obtained from the laboratory test. In the figure the pressiometer pressure is presented as a function of the relative volume change. The latter quantity cannot directly be obtained from PLAXIS and was calculated from the original radius R0 and the lateral expansion ux of the pressiometer: ∆V V0 = 2 ( R0 + u x )2 - R0 2 R0 Up to a pressure of 1600 kPa the results match quite well. Above 1600 kPa there is a sudden decrease in stiffness in the real test data, which cannot be explained. Nevertheless, the original set of parameters for the dense sand that were derived from triaxial testing also seem to match the pressiometer data quite well. 2500 pressure [kPa] 2000 1500 1000 500 0 0 0.1 0.2 dV/V0 0.3 0.4 Plaxis Test data Figure 8.15 Comparison of numerical results and pressiometer test data Conclusion The above results indicate that by use of the Hardening Soil model it is possible to simulate different laboratory tests with different stress paths. This cannot be obtained with simple models such as Mohr-Coulomb without changing input parameters. Hence, the parameters in the Hardening-Soil model are consistent and more or less independent from the particular stress path. This makes the HS model a powerful and an accurate model, which can be used in many applications. 8.3 SSC MODEL: RESPONSE IN ONE-DIMENSIONAL COMPRESSION TEST In this section the behaviour of the Soft-Soil-Creep model is illustrated on the basis of a one-dimensional compression test on clay. Two types of analysis are performed. First, the test is simulated assuming drained conditions in order to demonstrate the logarithmic stress-strain relationship and the logarithmic time-settlement behaviour on the long term (secondary compression). Second, the test is simulated more realistically by including undrained conditions and consolidation. Since the consolidation process depends on the drainage length, it is important to use actual dimensions of the test set-up. In this case an axisymmetric configuration with specimen height of 0.01 m, Figure 8.16, is used. The material parameters are shown in Table 8.3. The parameter values are selected arbitrarily, but they are realistic for normally consolidated clay. The vertical preconsolidation stress is fixed at 50 kPa (POP = 50 kPa). Figure 8.16 One-dimensional compression test Drained analysis In the first analysis successive plastic loading steps are applied using drained conditions. The load is doubled in every step using Staged construction with time increments of 1 day. After the last loading step an additional creep period of 100 days is applied. The calculation scheme is listed in Table 8.4. All calculations are performed with a tolerance of 1%. Table 8.3 Soft-Soil-Creep model parameters for one-dimensional compression test Parameter Unit weight Permeability Modified compression index Modified swelling index Secondary compression index Poisson's ratio Cohesion Friction angle Dilatancy angle Coeffient of lateral stress Symbol Value 19 0.0001 0.10 0.02 0.005 0.15 1.0 30 0.0 0.5 Unit kN/m3 m/day kN/m2 ° ° - γ kx, ky λ* κ* µ* νur c ϕ ψ K0nc Table 8.4 Calculation scheme for the drained case Phase 1 2 3 4 5 6 7 8 Calculation type Plastic Plastic Plastic Plastic Plastic Plastic Plastic Plastic Load input: Staged construction [kPa] 10 20 40 80 160 320 640 640 Time increment [day] 1 1 1 1 1 1 1 100 End time [day] 1 2 3 4 5 6 7 107 Undrained analysis In the second analysis the loading steps are instantaneously applied using undrained conditions. After each loading step a consolidation of 1 day is applied to let the excess pore pressures fully dissipate. After the last loading step, an additional creep period of 100 days is again introduced. The calculation scheme for this analysis is listed in Table 8.5. All calculations are performed with a reduced tolerance of 1%. Table 8.5 Calculation scheme for second analysis Phase 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Calculation type Plastic Consolidation Plastic Consolidation Plastic Consolidation Plastic Consolidation Plastic Consolidation Plastic Consolidation Plastic Consolidation Consolidation Loading input: Staged construction [kPa] 10 10 20 20 40 40 80 80 160 160 320 320 640 640 640 Time increment [day] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 100 End time [day] 0 1 1 2 2 3 3 4 4 5 5 6 6 7 107 C) As 'A' using Updated Mesh calculation Figure 8.18 Time-settlement curve of oedometer test with Soft-Soil-Creep model.Figure 8. A) Transient loading with doubling of loading within one day. after . B) Instantaneous loading with doubling of load at the beginning of a new day.17 Load-settlement curve of oedometer test with Soft-Soil-Creep model. A) Transient loading with doubling of loading within one day. It can be seen that. B) Instantaneous loading with doubling of load at the beginning of a new day Figure 8.17 shows the load-settlement curves of both analyses. 19 Stress states at a vertical stress level of -80 kPa. the lateral stress is determined by K0NC.influence of the preconsolidation stress can clearly be seen. From the slope of the primary loading line one can back-calculate the modified compression index λ* = ∆ε1 / ln((σ1+∆σ1)/σ1) ≈ 0.10. Left. much less than the decrease of the .005 (with t0 = 1 day). From the last part of the curve one can back-calculate the secondary compression index µ* = ∆ε1 / ln(∆t/t0) ≈ 0. As expected the horizontal stress is found to be approximately -40 kPa (corresponding to K0NC = 0. although the transition between reloading and primary loading is not as sharp as when using the Soft-Soil model. During primary loading. the lateral stress decreases much less than the vertical stress.5). after unloading from -640 kPa σxx’ ≈ -220 kPa Figure 8. For an axial strain of 30% one would normally use an Updated Mesh analysis. appropriate for normally consolidated soil. (∆σ′xx = 100 kPa). Figure 8. In fact. The plot in the right hand side shows the final situation after unloading down to -80 kPa. If. both at a vertical stress level of 80 kPa. Note that 1 mm settlement corresponds to ε1 = 10%. the results presented here are more realistic. i. The transition is indeed around 50 kPa.4) in which the vertical stress is reduced to 80 kPa. after primary loading σxx’ ≈ -40 kPa. the SoftSoil-Creep model would have been used in an Updated Mesh analysis with axial strains over 15% one would observe a stiffening effect as indicated by line C in Figure 8. To show these effects the calculation is continued after with a new drained unloading phase that starts from phase 7 (see Table 8.17. During unloading. Figure 8.18 shows the time-settlement curves of the drained and the undrained analyses. however.e. Right. Another interesting phenomenon is the development of lateral stresses. so that the ratio σ′xx / σ′yy increases.19 shows the stress state for two different calculation phases. The plot in the left hand side shows the stress state after primary loading. In this case the horizontal stress is decreased from -320 kPa to approximately -220 kPa. which has not been done in this simple analysis. vertical stress (∆σ′yy = 560 kPa).2) It is easy to verify that the results correspond to Poisson's ratio νur = 0. The model parameters are obtained from test results on Haney Clay (see Chapter 6) and are listed in Table 8. prescribed displacements are also applied.5 mm2).0001 Unit kN/m2 ° ° m/day λ κ* µ* νur c ϕ ψ K0nc Kx . In addition to isotropic loading. During sudden unloading in a one-dimensional compression test. The specimen surfaces (top and right hand side in Figure 8. 8.15 as listed in Table 8. Figure 8.4 SSC MODEL: UNDRAINED TRIAXIAL TESTS AT DIFFERENT LOADING RATES In this section the Soft-Soil-Creep model is utilised for the simulation of clay in an undrained triaxial test at different strain rates.20) are assumed drained whereas the other boundaries are assumed closed. here. the behaviour is purely elastic.0 32 0. Hence.20.1. a situation where σ′xx is larger than σ′yy is obtained. horizontal and vertical loads (both System . ky Modelling of the triaxial test is as described in Section 8.ν ur (8. Both types of loading are simulated using the Staged construction option. Thus.3.6 Soft-Soil-Creep model parameters for Haney clay Parameter Modified compression index Modified swelling index Secondary compression index Poisson's ratio Cohesion Friction angle Dilatancy angle Coeffient of lateral stress Permeability * Symbol Value 0.61 0.15 0.0 0.105 0. Table 8. However.5 x 17. the real dimension of the test set-up is simulated (17.6. the ratio of the horizontal and vertical stress increments can be determined as: ∆ σ ’ xx ν ur = ∆ σ ’ yy 1 .004 0.016 0. During isotropic loading. Figure 8.01 8. Initial configuration.00 0.0021 Time increment [day] 0.01 0.0556 0.15%/minute) and 0. Table 8.01 0. Right.00094%/minute).1 mm) is applied in 8. the displacements are reset to zero.00 0. As such a total of 12% axial strain (2.10%/minute) respectively.0021 0.20 Modelling of triaxial test on Haney clay.7. Each of the prescribed displacement loading phase starts from the end of the isotropic loading phase. 0.A) are applied.7 Loading scheme for triaxial tests at different loading rates Phase 1 2 3 4 5 6 7 8 9 10 (start from 8) 11 (start from 8) Calculation Plastic Consolidation Plastic Consolidation Plastic Consolidation Plastic Consolidation Plastic Plastic Plastic Applied Load [kPa] 65 65 130 130 260 260 520 520 520 520 520 Applied displacement [mm] Inactive Inactive Inactive Inactive Inactive Inactive Inactive Inactive 0.0021 0.11 After the isotropic loading phases.865 0.865 days (0.00 0.00 0. configuration for phase 9 . The calculation phases for isotropic loading consist of undrained plastic and consolidation analyses.0556 days (0.01 0. Left. The calculation scheme is listed in Table 8. Rate of loading is simulated by applying prescribed displacements at different velocities.00758 days (1.00758 . The vertical load is deactivated and the prescribed displacement is activated. For higher strain rates there is a smaller reduction of the mean effective stress. It can be seen that the shear strength highly depends on the strain rate.01) . 0. Figure 8. the higher strain rate the higher the shear strength. Figure 8. This is due to the fact that points close to draining boundaries consolidate faster than points at a larger distance.22 p-q stress paths for different rates of straining for a point at position (0.The computational results are presented in Figure 8.22 shows the p-q stress paths from the prescribed displacement loading phases. because of the inhomogeneous (excess) pore pressure distribution.22. which allows for a larger ultimate deviatoric stress. Figure 8.21 and Figure 8.21 Average deviatoric stress versus axial strain for different rates of straining Figure 8.01 . It should be noted that the stress state is not homogeneous at all.21 shows the stress-strain curves of the prescribed displacement loading phases. The parameters are presented in Table 8.0 kPa 0.8 SS Model parameters for isotropic compression test Modified compression index Modified swelling index Poisson's ratio Friction angle Cohesion Normally consolidated K0 λ* κ* νur ϕ c K0NC 0. The parameters of the Soft-Soil model are chosen arbitrarily.9. Then.e.5 From a stress-free state. the isotropic pressure is increased to p′ = 1000 kPa. the sample is loaded up to p′ = 10000 kPa.. and hence. the preconsolidation stress is equivalent to the current stateof-stress.8. In the last loading path. The vertical load (A) and the horizontal load (B) are simultaneously applied to the same level so that a fully isotropic stress state occurs.1.02 0. This loading path is denoted as 'primary loading'. Table 8. the sample is isotropically 'unloaded' to p′ = 100 kPa. For this purpose the test set up is simulated as that presented in Figure 8.9 Calculation phases for isotropic compression test on clay Stage 0 1 2 3 4 Initial situation Primary loading Unloading Reloading Primary loading p0 = 100 kPa p1 = 1000 kPa p2 = 100 kPa p3 = 1000 kPa Initial stress Final stress p0 = 100 kPa p1 = 1000 kPa p2 = 100 kPa p3 = 1000 kPa p4 = 10000 kPa . it consists of two parts: the part of the loading path for which p′ < 1000 kPa is referred to as 'reloading'. and the part of the loading path for p′ > 1000 kPa consists of further primary loading.5 SS MODEL: RESPONSE IN ISOTROPIC COMPRESSION TEST In this Section it will be demonstrated that the Soft-Soil model obeys a logarithmic relationship between the volumetric strain and the mean stress in isotropic compression. Finally. Table 8. the maximum preload of 1000 kPa is exceeded. The calculation phases are indicated in Table 8. i. the model is isotropically loaded to a mean stress of p′ = 100 kPa.15 30° 1. after which the displacements are reset to zero.10 0. As a result. After that. but the values are realistic for normally consolidated clay. the material becomes 'normally consolidated'.8. 102 1 0 ln(1000 / 100 ) ln p / p ( ) .235 εv2 = .The computational results are presented in Figure 8. Phase 1 λ* = − 1 0 0.0. (7. The plot shows two straight lines. the parameters λ* and κ* can be backcalculated using Eqs.2). which indicates that there is indeed a logarithmic relation for loading and unloading.0. which shows the relation between the vertical strain εyy and the vertical stress σ′yy.188 εv3 = .235 εv1 = . The vertical strain is 1/3 of the volumetric strain. Table 8. Figure 8.000 εv1 = .0.235 εv2 = .000 εv0 = 0.235 εv − εv = = 0. The volumetric strains obtained from the calculation are given in Table 8.0.0. p′.471 From these strains and corresponding stresses.235 εv4 = .188 εv3 = . εv. and the vertical stress is equal to the mean stress.23 Results of isotropic compression test The latter quantity is plotted on a logarithmic scale.0.10.1) and (7.10 Volumetric strains from various calculation phases Phase 0 1 2 3 4 Initial strain Final strain εv0 = 0.23.0. 471 − 0.020 ln(1000 / 100) ln p3 / p 2 ) Phase 4 4 3 ε v − ε v = 0.6 SUBMERGED CONSTRUCTION OF AN EXCAVATION WITH HS MODEL In this example a particular advantage of the Hardening-Soil model is demonstrated.188 = 0. Figure 8. An example of an application with this model is described in Section 8.235 − 0.8.102 4 ln(10000 / 1000) ln p / p 3 ) The back-calculated values correspond to the input values as given in Table 8.Phase 2 κ* = − κ* = − λ* = − 2 1 ε v − ε v = 0.235 = 0.235 = 0. Such behaviour is included in the Soft-Soil-Creep model.020 ln p2 / p1 ln (100 / 1000) ( ( ( ) Phase 3 3 2 ε v − ε v = 0.188 − 0. 8.7. namely the distinction between loading and unloading stiffness.24 Geometry model of submerged excavation . This feature becomes particularly important in quasi unloading problems like excavations and tunnels. Note that the Soft-Soil model does not include time effects such as in the secondary compression. 8 0. In order to show that the results can be improved by using the Hardening-Soil model. which is the default setting in PLAXIS. Figure 8.0 32 2 0.2 0.9 Sand layer 17 / 20 30000 90000 30000 1.24 but slightly simplified.11 HS model parameters for the two layers in the excavation project Parameter Unit weight above/below phreatic level E50ref (pref = 100 kPa) Eurref (pref = 100 kPa) Eoedref (pref = 100 kPa) Cohesion c Friction angle ϕ Dilatancy angleψ Poisson's ratio νur Power m K0nc Tensile strength Failure ratio Clay layer 16 / 18 8000 24000 4000 5. which pushes the soil up. The model parameters are listed in Table 8.In the submerged excavation example. .47 0.25. The deformed mesh clearly shows that there is a limited heave at the bottom of the excavation.11. a similar example is used here again. Most of the deformation is caused by the horizontal movement of the diaphragm wall. the excavation is executed in two phases. The two soil layers are modelled here with the Hardening-Soil model instead of the Mohr-Coulomb model.5 0. The geometry model is similar to the one used in Lesson 2. The results of the excavation are shown in Figure 8. in contrast to the Mohr-Coulomb model. The difference can be explained by the fact that. Table 8. The unloading stiffness is set here as three times higher than the loading stiffness.0 25 0 0.0 0. The vertical heave at the bottom further away from the wall is very low as compared to the results presented in the Tutorial Manual. the Hardening-Soil model distinguishes between loading and unloading stiffness.2 0. presented in the Tutorial Manual.0 0.9 Unit kN/m3 kN/m2 kN/m2 kN/m2 kN/m2 ° ° kN/m2 - After the generation of the initial pore pressures and effective stresses. the heave at the bottom of the excavation is unrealistically large.5 0. One of these features is the reduction of the mean effective stress during undrained loading due to compaction of the soil.25 Deformed mesh after excavation 8.7 ROAD EMBANKMENT CONSTRUCTION WITH THE SSC MODEL This example demonstrates some of the features of the Soft-Soil-Creep model in simulating soils for engineering problems. This feature becomes particularly important in embankment construction projects.Figure 8. in the first part of Lesson 5 of the Tutorial Manual (construction of a road embankment) the safety factor was relatively low during construction. For example. A B Figure 8.26 Geometry model of road embankment project . since it highly influences the stability of the embankment during construction. Please note that if for some reason a normal calculation phase (such as Staged construction or phi-c reduction) is needed after an Updated mesh phase. is reanalysed here. The model parameters for this layer are listed in Table 8.4 Unit kN/m3 kN/m3 m/day m/day kN/m2 ° ° - The calculation phases are shown in Table 8. To illustrate these effects.26. the safety factor will become even lower.59 1. The fourth phase is conducted using the Minimum pore pressure loading input and the Safety analysis phase is conducted using the Incremental multipliers loading input.When using the Soft-Soil-Creep model for the clay layer with effective strength properties similar to those of the Mohr-Coulomb model. This is attributed to the fact that with this option the phi-c reduction analysis will produce an infinite safety. the Updated mesh option should not be used. The peat layer and the sand embankment are simulated by means of Mohr-Coulomb using the same parameters as given in Lesson 5 of the Tutorial Manual. The first three phases are conducted using Staged construction. Figure 8. It is worth mentioning here that if a Safety factor analysis is required.007 0.12 Soft-Soil-Creep model parameters for undrained clay layer Parameter Unit soil weight above phreatic level γunsat Unit soil weight below phreatic level γsat Horizontal permeability kx Vertical permeability ky Modified compression index λ* Modified swelling index κ* Secondary compression index µ* Poisson's ratio νur Cohesion c Friction angle ϕ Dilatancy angle ψ Coeffient of lateral stress K0NC OCR Clay layer 15 18 1 ·10-4 1 ·10-4 0. the embankment in Lesson 5. the displacement must be reset to zero. . Table 8. the clay layer is simulated by means of the Soft-Soil-Creep model.035 0. In this section a similar geometry model as that of Lesson 5 is used.15 2 24 0 0. however.12.002 0.13. .1 Time interval 5 day 200 days 5 days Figure 8. the figure shows the results of the Safety factor analysis when using the Mohr-Coulomb model (see the Tutorial Manual). while by the Mohr-Coulomb model is 1.27 Safety factor analysis using MC model and SSC model. Safety factor 1 .Table 8. Ultimate time Consolidation. the calculation of the other phases is not possible because the structure has already reached to its failure state. Ultimate time Consolidation.02 1 . 11 1 .11. = 1 day Msf = 0. Ultimate time Consolidation. First time incr. This difference is due to the reduction of the mean effective stress as a result of the accumulation of the irreversible compaction (volume creep strain) in the SSC model. Also.00 0 MC m odel SSC m odel Displacem ent [m ] Figure 8.1 4 1 .08 1 .27 shows the results of the Safety factor analysis after the construction of the first layer of the embankment (after 5 days). Minimum pore pressure Safety analysis Loading input: 1st embankment part Desired maximum 50.05 1 .01. It can be seen that the safety factor produced by the Soft Soil Creep model is 1. First time incr.13 Overview of calculation phases Phase 1 2 3 4 5 Start from 0 1 2 3 1 Type Consolidation. Apparently due to this low safety factor. = 1 day 2nd embankment part Desired maximum 30. 2 MC model 0.Displacement [m] -0. .4 0.4 MC model -0.1 0 0 100 200 300 400 500 600 Time [day] (a) Displacement [m] 0. (b) horizontal displacements at B. MC model (a) vertical displacements at A.28 SSC model vs.5 SSC model -0.5 SSC model 0.1 0 0 100 200 300 400 500 600 Time [day] (b) Figure 8.2 -0.3 -0.3 0. In order to study the influence of the SSC model on the response of the embankment.28(a) shows the vertical displacements at point A and Figure 8. stems from the fact that no undrained strength properties can be specified (only c' and ϕ') and that the numerical procedure becomes more complicated (less robust) when soil failure is approached. The results are compared with those obtained by using Mohr-Coulomb model. (see Figure 8. . The Soft-Soil-Creep model does include the effect of reducing mean effective stress during undrained loading.29 Tendency towards failure mechanism after the construction of second part of the embankment Conclusion The reduction of mean effective stress during undrained loading is a known phenomenon in soil engineering. It can be seen that the SSC model exhibits little differences in the vertical displacement but much larger differences in the horizontal displacement. Figure 8. This phenomenon has a negative influence on the strength and stability of the soil structure. In such cases it is better to use undrained strength properties in the Mohr-Coulomb model (c = cu and ϕ = 0). this effect is not taken into account and leads to an over prediction of the stability when using effective strength properties.26). like the Mohr-Coulomb model. Figure 8. an Updated mesh analysis is conducted. including time-dependent behaviour (secondary compression and consolidation). When using simple soil models. The later can be attributed to the fact that the material here is close to failure due to the creep effect.28(b) shows the horizontal displacements at point B. This model gives a more realistic prediction of soft-soil behaviour. Figure 8. however. The disadvantage of this model.29 shows the tendency towards failure after the construction of the second layer. σ ij . then compiled as a Dynamic Link Library (DLL) and then added to the PLAXIS program directory. which is included in the program CD.1 INTRODUCTION PLAXIS Version 8 has a facility for user-defined (UD) soil models.κ σ tij . IMPLEMENTATION PROGRAM OF UD MODELS IN CALCULATIONS 9.2 The PLAXIS calculations program has been designed to allow for user-defined soil models. Such models must be programmed in FORTRAN (or another programming language). a UD subroutine based on the Drucker-Prager material model is provided in the user-defined soil models directory. > Please note that the PLAXIS organization cannot be held responsible for any malfunctioning or wrong results due to the implementation and/or use of user-defined soil models. ∆ t As an example. In this section. κ t t +∆t t + ∆t current stresses and state variables previous stresses and state variables strain and time increments ∆ ε ij . There are mainly four tasks (functionalities) to be performed in the calculations program: • • • Initialisation of state variables Calculation of constitutive stresses (stresses computed from the material model at certain step) Creation of effective material stiffness matrix . This facility allows users to implement a wide range of constitutive soil models (stress-strain-time relationship) in PLAXIS. In the material data base of the PLAXIS input program. In principle the user provides information about the current stresses and state variables and PLAXIS provides information about the previous ones and also the strain and time increments.9 USER-DEFINED SOIL MODELS 9. a step-by-step description on how a user-defined soil model can be formed and utilised in PLAXIS is presented. the required model parameters can be entered in the material data sets. 12 for 15-noded elements) Global coordinates of current stress point Time at the start of the current step Time increment of current step Array(1. In this subroutine more than one UD soil model can be defined.. 4 = Return the number of state variables. σ'yy0. σ'zx0). NonSym. dEps. To create a UD soil model. iMod.50) with User-defined model parameters for the current stress point Array(1. Swp0. the User_Mod subroutine must have the following structure: Subroutine User_Mod (IDTask. In 2D calculations σyz and σzx should be zero. 5 = Return matrix attributes (NonSym. the calculation program calls the corresponding task from the subroutine User_Mod. iTimeDep). Previous excess pore pressure of the current stress point iMod IsUndr iStep iter Iel Int X.• Creation of elastic material stiffness matrix These main tasks (and other tasks) have to be defined by the user in a subroutine called 'User_Mod'. 2 = Calculate constitutive stresses. Time0. StVar0. iPrjLen. ipl... Sig. iStrsDep. σ'xy0. iPrjDir.Z Time0 dTime Props Sig0 = = = = = = = = = = = Swp0 = . iStrsDep. Y. σ'yz0. Sig0. σ'zz0. StVar.) Drained condition (IsUndr = 0) or undrained condition (IsUndr = 1) Current calculation step number Current iteration number Current element number Current local stress point number (1. iTimeDep. 6 = Create elastic material stiffness matrix) User-defined soil model number (This option allows for more than one UD model. dTime.6) with previous (= at the start of the current step) effective stress components of the current stress point (σ'xx0. Props.. Bulk_W. iTang. D. If a UD soil model is used in an application. IsUndr.Y. Iel. Z. Swp. nStat. iAbort) where: IDTask = Identification of the task (1 = Initialise state variables. iStep. or 1.3 for 6-noded elements. iTer. X. 3 = Create effective material stiffness matrix. Int. . σ'zx) Resulting excess pore pressure of the current stress point Array(1.6) Bulk modulus of water for the current stress point (for undrained calculations and consolidation) Array (1. ∆γzx) Effective material stiffness matrix of the current stress point (1. ∆εzz. to be used in a full NewtonRaphson iteration process (iTang = 1) or not (iTang = 0).StVar0 dEps D Bulk_W Sig Swp StVar ipl = = = = = = = = Array(1. 2 = Tension cut-off point.nStat) with resulting values of state variables for the current stress point Plasticity indicator: 0 = no plasticity. 3 = Cap hardening point.. 1 = Mohr-Coulomb (failure) point. iStrsDep = iTimeDep = iTang = iPrjDir iPrjLen iAbort = = = ..6.6) with strain increments of the current stress point in the current step (∆εxx. ∆γyz. Parameter indicating whether the material stiffness matrix is stress-dependent (iStrsDep = 1) or not (iStrsDep = 0). ∆εyy..6) with resulting constitutive stresses of the current stress point (σ'xx. 5 = Friction hardening point. Parameter indicating whether the material stiffness matrix is time-dependent (iTimeDep = 1) or not (iTimeDep = 0). σ'yy. σ'zz. σ'yz. σ'xy. Parameter indicating whether the material stiffness matrix is a tangent stiffness matrix.. ∆γxy. 4 = Cap friction point. 1.nStat) with previous values of state variables of the current stress point Array(1. nStat NonSym = = Number of state variables (unlimited) Parameter indicating whether the material stiffness matrix is non-symmetric (NonSym = 1) or not (NonSym = 0) (required for matrix storage and solution).. Project directory (for debugging purposes) Length of project directory name (for debugging purposes) Parameter forcing the calculation to stop (iAbort = 1). e. The user subroutine should contain program code for listing the tasks and output parameters (IDTask = 1 to 6). The values of these parameters are provided by PLAXIS and can be used within the subroutine. indicated by the UD model number iMod. distinction should be made between different models. The parameters D to iTang and iAbort are output parameters. the User_Mod subroutine must have the following structure (here specified in pseudo code): Case IDTask of 1 Begin { Initialise state variables StVar0 } End 2 Begin { Calculate constitutive stresses Sig (and Swp) } End 3 Begin { Create effective material stiffness matrix D } End 4 Begin { Return the number of state variables nStat } End 5 Begin { Return matrix attributes NonSym. iTimeDep} End 6 Begin { Create elastic material stiffness matrix De } End End Case If more than one UD model is considered. In case IDTask = 1. . 'Previous' means 'at the start of the current step'. After the declaration of variables. i. The values of these parameters are to be determined by the user.In the above. 'increment' means 'the total contribution within the current step' and not per iteration. parameters beginning with the characters A-H and O-Z are double (8-byte) floating point values and the remaining parameters are 4-byte integer values. iStrsDep. In the terminology of the above parameters it is assumed that the standard type of parameters is used. StVar0 becomes output parameter. The parameters IDTask to dEps and iPrjDir and iPrjLen are input parameters. These input parameters should not be modified (except for StVar0 in case IDTask = 1). which is equal to the value at the end of the previous step. Let us consider a simple example using a linear elastic D-matrix as created under IDTask = 3. used in hardening models to indicate the current position of the yield loci. For this case the initial value has to be calculated based on the actual conditions (actual stress state) at the beginning of the step. because the initialization of this user-defined variable is not considered in the K0-procedure. if the previous calculation step does not contain information on the state variables (for example in the very first calculation step). the situation where the first state variable is the minimum mean effective stress. Sig. Sig0.e. part 2 of the user subroutine may look like: . can directly be calculated from the initial stresses. The resulting value of the state variable in the previous step. Within a continuous calculation phase. dEps: Sig[i] = Sig0[i] + ∑(D[i.0 StVar0[1] = Min(StVar0[1]. In this case. the StVar0 array would contain zeros. for example. for example. However. part 1 of the user subroutine may look like: 1 Begin { Initialise state variables StVar0} p = (Sig0[1] + Sig0[2] + Sig0[3]) / 3. state variables are automatically transferred from one calculation step to another. In this case the stress components. i. it is necessary to know about the initial value of the state variables.p) End Calculate constitutive stresses (IDTask = 2) This task constitutes the main part of the user subroutine in which the stress integration and correction are performed according to the user-defined soil model formulation.Initialise state variables (IDTask = 1) State variables (also called the hardening parameters) are. If the initial stresses have been generated using the K0procedure. p' (considering that compression is negative). The update of state variables is considered in the calculation of constitutive stresses based on the previous value of the state variables and the new stress state. In this case. When starting a new calculation phase. then the initial effective stresses are non-zero. StVar. In this case it is not necessary to modify the StVar0 array. the material stiffness matrix. Consider. and the strain increments. is stored in the output files and automatically used as the initial value in the current step. j]*dEps[j]). but the initial value of the state variable is zero. StVar0. Hence. the initial value of the state variables is read from the output file of the previous calculation step and put in the StVar0 array. the value at the beginning of the calculation step. D. 6] = End effective material stiffness matrix D } Props[1] Props[2] 0.j]*dEps[j] End for {j} End for {i} End Create effective material stiffness matrix (IDTask = 3) The material stiffness matrix. may be a matrix containing only the elastic components of the stress-strain relationship (as it is the case for the existing soil models in PLAXIS).5] = D[6. ν.3] = D[2.0+v) 2*G/(1.2] = D[3.2] = D[2. and Poisson's ratio.5*E/(1.. D. respectively. Let us consider the very simple example of Hooke's law of isotropic linear elasticity. There are only two model parameters involved: Young's modulus. in position 1 and 2 of the model parameters array.3] = D[4. These parameters are stored.0-2*v) { make sure that v < 0.1] = D[1.5 !! } Fac*(1-v) Fac*v Term1 Term2 Term2 Term2 Term1 Term2 Term2 Term2 Term1 G G G .1] = D[2. In this case.2] = D[1. E.1] = D[3. or the full elastoplastic material stiffness matrix (tangent stiffness matrix).50). Props(1.2 Begin { Calculate constitutive stresses Sig (and Swp) } For i=1 to 6 do Sig[i] = Sig0[i] For j=1 to 6 do Sig[i] = Sig[i] + D[i.3] = D[3.4] = D[5. part 3 of the user subroutine may look like: 3 Begin { Create E = v = G = Fac = Term1 = Term2 = D[1. for example in the case of nonassociated plasticity. After calling the user subroutine with IDTask = 3 and IsUndr = 1. the user subroutine should look like: 4 Begin { Return the number of state variables nStat } nStat = 1 End Return matrix attributes (IDTask = 5) The material stiffness matrix may be stress-dependent (such as in the Hardening Soil model) or time-dependent (such as in the Soft Soil Creep model). For iStrsDep = 1 the global stiffness matrix is created and decomposed at the beginning of each calculation step based on the actual stress state (modified Newton-Raphson . The last part of the user subroutine is used to initialize the matrix attributes in order to update and store the global stiffness matrix properly during the calculation process. In this case the user subroutine may be written as: 5 Begin { Return matrix attributes NonSym. } { iTimeDep. it is a good habit to explicitly define zero terms as well.(By default... For the simple example of Hooke’s law.3] = D[i. i. the number of state variables. iStrsDep.3. iTang } NonSym = 0 iStrsDep = 0 iTimeDep = 0 iTang = 0 End For NonSym = 0 only half of the global stiffness matrix is stored using a profile structure. whereas for Nonsym = 1 the full matrix profile is stored. In the case of just a single state parameter. Return the number of state variables (IDTask = 4) This part of the user subroutine returns the parameter nStat. When using a tangent stiffness matrix. j=1. so the remaining terms are still zero.. D will be initialized to zero.3]).e. If Bulk_W is not specified.) If undrained behaviour is considered (IsUndr = 1). as described earlier..3. j=1. PLAXIS will give it a default value of 100*Avg(D[i=1. then a bulk stiffness for water (Bulk_W) must be specified at the end of part 3. the matrix is symmetric and neither stress. j]+ Bulk_W. however. PLAXIS will automatically add the stiffness of the water to the material stiffness matrix D such that: D[i=1.nor time-dependent. the matrix may even be non-symmetric. For iTang = 1 the global stiffness matrix is created and decomposed at the beginning of each iteration based on the actual stress state (full Newton-Raphson procedure. The currently activated load is the load that is being activated in the current calculation phase. a stricter equilibrium condition must be adopted. whereas the previously activated load is the load that has been activated in previous calculation phases and that is still active in the current phase. Part 6 of the user subroutine looks very similar to part 3. The CSP parameter is defined as: CSP = Total elastic work Total work The elastic material stiffness matrix is required to calculate the total elastic work in the definition of the CSP. However in the case that an elastoplastic or tangent matrix was used for the effective stiffness matrix. Create elastic material stiffness matrix (IDTask = 6) The elastic material stiffness matrix. The reason that an elastic material stiffness matrix is required is because PLAXIS calculates the current relative global stiffness of the finite element model as a whole (CSP = Current Stiffness Parameter). The global error is defined as: Global error = unbalance force currently activated load + CSP ⋅ previously activated load The unbalance force is the difference between the external forces and the internal reactions. Using the above definition for the global error in combination with a fixed tolerated error results in an improved equilibrium situation when plasticity increases or failure is approached. this matrix may just be adopted here.For iTimeDep = 1 the global stiffness matrix is created and decomposed every time when the time step changes. to be used in combination with iStrsDep=1). then the matrix to be created here should only contain the elastic components. safety factor or bearing capacity. The CSP equals unity if all the material is elastic whereas it gradually reduces to zero when failure is approached. but in order to accurately calculate failure state. De. It should be noted that the same variable D is used to . is the elastic part of the effective material stiffness matrix as described earlier. The idea is that a small out-of-balance is not a problem when a situation is mostly elastic. In the case that the effective material stiffness matrix was taken to be the elastic stiffness matrix. except that only elastic components are considered here. The CSP parameter is used in the calculation of the global error. . The available subroutines may be called in by User_Mod subroutine to shorten the code.lib) and in the source code (to be included in the file with the user subroutine).): DLL_Export User_Mod Using Digital Visual Fortran: !DEC$ ATTRIBUTES DLLExport :: User_Mod In order to compile the USRMOD... a number of FORTRAN subroutines and functions for vector and matrix operations are available in PLAXIS in specific compiler libraries (LFUsrLib.store the elastic material stiffness matrix. This file should be placed in the PLAXIS program directory. The following statement has to be inserted in the subroutine just after the declaration of variables: • • Using Lahey Fortran (LF90.. It is supposed that the user subroutine User_Mod is contained in the file USRMOD. LF95.FOR DFUsrLib.. Compiling user subroutine The user subroutine User_Mod has to be compiled into a DLL file using appropriate compiler.FOR -lib LFUsrLib -ml LF90 Using Digital Visual Fortran: DF /winapp USRMOD.lib or DFUsrLib.DLL file will be created.2] = D[1. thereafter it can be used together with the existing PLAXIS .FOR into a DLL file. After creating the user subroutine User_Mod.lib /dll In all cases USRMOD. . the following command must be executed: • • • Using Lahey Fortran 90: LF90 -win -dll USRMOD. 6 Begin { Create elastic material stiffness matrix D } D[1.FOR. a command must be included to export data to the DLL.6] = End Using predefined subroutines from libraries In order to simplify the creation of user subroutines. D[6. Note that the compiler must have the option for compiling DLL files. whereas in Part 3 this variable is used to store the effective material stiffness matrix.. An overview of the available subroutines is given in Appendix B.FOR -lib LFUsrLib Using Lahey Fortran 95: LF95 -win -dll USRMOD. Below are examples given for two different FORTRAN compilers.3] = .1] = D[1. In order to be able to effectively debug the user subroutine.1. usually some time is spent to 'debug' earlier written source code. the availably written subroutines (see Appendix B). A user-defined model can be selected from the Material model combo box in the General tab sheet.DLL file. for example. In Appendix C a suggestion on how this can be done is given. data can be written to this file from within the user subroutine. Such a 'debug-file' is not automatically available and has to be created in the user subroutine. (a) . Once the UD model is used. In fact. After the debug-file is created.3 INPUT OF UD MODEL PARAMETERS VIA USER-INTERFACE Input of the model parameters for user-defined soil models can be done using PLAXIS material data base. a window appears with three tab sheets: General. This can be done by using.EXE).calculations program (PLASW. PLAXIS will execute the commands as listed in the USRMOD. 9. the procedure is very similar to the input of parameters for the existing PLAXIS models. there should be a possibility for the user to write any kind of data to a file. Debugging possibilities When making computer programs. Figure 9. When creating a new material data set for soil and interfaces in the material data base. Parameters. Interface. In that case the next section can be skipped. all parameters as required for a particular UD model and their corresponding input unit must be specified in the table. l = unit of length. Defining a new set of UD model parameters When UD model has to be used in a project. In this way. The set of model parameters can be completed by pressing <OK> button. the corresponding set of the model parameters has to be defined. it is strongly advised to define the parameter units in these basic units (f = unit of force.Subsequently.2 shows the parameters and units for the Hooke's law example described above. the specific model parameters can be entered in the Parameters tab sheet. If the parameter set for a particular model has already been defined.1. Each UD model has its own set of model parameters. a maximum of 50 parameters can be entered. Figure 9. The ID parameter corresponds to the iMod parameter in the User_Mod subroutine.2. Powers of units can be indicated after using the hat (^) sign. after which the name of the model should be specified in the Model field. Figure 9.(b) Figure 9. The numbering of parameters corresponds to the Props array in the User_Mod subroutine. defined by the Model parameter. Then the UD model number must be selected correctly from the ID combo box. Units may be entered as common abbreviations. . Selection of user-defined soil models (a) and input of parameters (b) After inputting general properties. but in order to guarantee consistency with the basic units as defined in the General Settings. it is available in the combo box. t = unit of time). This can be done by pressing the <New> button. 2 Inputting of the UD model parameters After completing the set of model parameters it appears by its name in the Model combo box. Figure 9. Figure 9.Figure 9.3 Inputting model parameter values .3. 50) in the User_Mod subroutine. For user-defined soil models the interface tab sheet is slightly ref . . the interface shear strength is directly given in strength parameters instead of using a factor relating the interface shear strength to the soil shear strength. the data sets can be assigned to the corresponding soil clusters. this tab sheet contains the Rinter parameter and the selection of the type of interface regarding permeability. contains the material data for interfaces. Figure 9.. ϕinter and ψinter. Eoed parameters cinter. and the interface strength different and contains the interface oedometer modulus.Existing sets may be deleted using the <Delete> button or modified using the <Edit> button.4. Normally. Figure 9. Be careful with deleting or modifying existing parameter sets. Interfaces The Interfaces tab sheet. Hence. since other projects may still use these parameter sets. The user-defined parameters are transmitted to the calculation program and appear for the appropriate stress points as Props(1. as it is the case in PLAXIS models.4 Interface tab sheet After having entered values for all parameters. in a similar way as for the existing material models in PLAXIS. . Pietruszczak). 2. T. 1. Vermeer. Quart. J. S.C. Soils and Foundations 22: 57-70. 157-165. 1: 103-107.. Bolton. Dissertation. Computer Methods in Applied Mechanics and Engineering 78: 48-72. Buisman. W. Borja. Geotechnique 29:469-480. Drucker. E. No. 211-214. R.. Symposium on Numerical Models in Geomechanics (eds.J. Localization limiters and numerical strategies for strain-softening materials. Balkema. (1994). Amsterdam. pp 349-362. Burland. (1985). 2. No. Borja.. Geomaterial Models and Numerical Analysis of Softening. Deformation of Soft Clay. (1967). R.... 15. (1967)..B. Brinkgreve. Proc. Rotterdam. Soil Mechanics and Plastic Analysis or Limit Design.. Geotechnique 17: 81-118. (1952). K. R.I. Oka.A. R. Mazars & Bazant). (1990). Cam-clay plasticity. France-US Workshop on Strain localization and size effect due to cracking and Damage (eds.. pp. Cambridge. pp. Dissertation. J.D. McGraw-Hill. 36. D. G..R.F. Appl. L. W. (Correspondence). Géotechnique. F. Seventh Rankine Lecture. Math. Brinkgreve. P. Chen. D. (1965). Vol. T. Mass. The Strength and Dilatancy of Sands. Proceedings 1st International Conference on Soil Mechanics and Foundation Engineering.. Vol. R. 65-78. Engineering geology of Norwegian normally-consolidated marine clays as related to settlements of buildings. S. London. part 1: implicit itegration of elasto-plastic constitutive relations. Vol. (1986). Bjerrum. Elsevier.J. Cambridge University. Kavaznjian.10 REFERENCES [1] [2] [3] Adachi.. Vol.. Prager. 557-565.B. 10. Vol. Pande. J. (1975). Burland. Géotechnique. (1989). Butterfield. Belytschko. (1936). P..H. Results of long duration settlement tests. On the use of Cam-Clay models. IV Int. Limit Analysis and Soil Plasticity..I... The Yielding and Dilation of Clay. Constitutive equation for normally consolidated clays based on elasto-viscoplasticity. Delft University of Technology. A natural compression law for soils (an advance on elog p’).N. Geotechnique 35: 283-298. [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] . The Mechanics of Soils.B.L. Lasry. (1979). pp.. Proc. Atkinson. (1982).. (1992). A constitutive model for the σ-ε-t behaviour of wet clays. M. Lee... Bransby. (1978). pp.B. Koiter. N. (1965). E. 165-221. P. 1. Soil Behaviour and Critical State Soil Mechanics.. T. The Mathematical Theory of Plasticity. Neher. Undrained Stress-Strain-Time Behaviour of Clays. 96. Div.E. pp. Prentice-Hall. (1998). (1960).. F. den Haan. (1977).. Am. Soil Compressibility as Determined by Oedometer and Triaxial Tests. Chang. The resistance concept applied to soils.-Y. Janbu.D. 500-527. Meth. pp. R. Geotechnique 22: 71-78. Vol.. and Found. Cambridge University Press. J. Leroueil.. (1963). thesis. Sneddon. Roy. Proc. General Theorems for Elastic-Plastic Solids. Prevost. 1. A Hyperbolic Stress Strain Formulation for Sands.L. Rowe.. Pan. H. The Stress-Dilatancy Relation for Static Equilibrium of an Assembly of Particles in Contact. P. In: Progress in Solid Mechanics (eds. Int. Int... Proc. Mayne. ICOSFE Brazil.. Proceedings of the 7h ICSMFE. Mexico City 1:191-196. 1. North-Holland.. 2. D. of the Soil Mech. London. Angles of Friction and Dilatancy of Sand. 145-151.[16] [17] [18] [19] [20] [21] [22] [23] Duncan. A. Québec. Thesis. pp. Foundations of Solid Mechanics. (1972). (1969).. P.. P. P. Soc.A. Vertical Compression of Soils. Manual on Estimating Soil Prperties for Foundation Design. Meth. 19-25.T. (1990). (1996). (1994). (1998). Geomech. (1963). (1990). van Langen. Fung. T. The consolidation of soils exhibiting creep under constant effective stress. Schanz. Oxford University Press.W. S.J... Formulation and application of a soil model that accounts for creep. Engng. H. Géotechnique 46. Hill). Garlanger. pp. USA. Amsterdam. Vol. Cornell University. Vol. Delft University. J. 289-324. (1990). Y. Habilitation. Ph. 1629-1653.A. H. Kulhawy. 29.. [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] ... Anal. Stuttgart Universität. Vol.M.. pp.C.-H. Journal of the Geotechnical Engineering Division GT12: 1245-1259. W. Quelques considérations sur le comportement des argiles sensibles. Nonlinear Analysis of Stress and Strain in Soil. ASCE J.. U. 579-598. pp. Muir Wood.. I. R. (1976).N...W. J. Janbu. No. ECSMFE Wiesbaden. Kondner. (1962). J. New Jersey.. Zur Modellierung des Mechanischen Verhaltens von Reibungsmaterialen. Vermeer. (1950).K. New York. C. Num. R. Numer. Hill. Schanz.. Laval University. J. pp. (1970).A. J. Ithaca. Vermeer. Automatic Step Size Correction for Non-Associated Plasticity Problems. Vol.. Vermeer. 269. Ernst & Sohn. Bonnier. Vermeer.. In: R. (1997). Y... Meth. P. I. P. pp. P. Rotterdam: 281-290. Smith. 103(GT7). Berlin. (1984). (1998). de Borst.A.G. D. Vermeer. Heron. Proc.. Numog 1997: 123-128. D.. Non-Associated Plasticity for Soils.E. P.. Ingenieur-Archiv 59: 221-236. (1998). Second Edition.A. Time-dependent behaviour of undisturbed clay.B. International Journal for Numerical and Analytical Methods in Geomechanics 15: 399-416. Vol 29.. [36] [37] [38] [39] [40] [41] [42] [43] [44] ..A. Vol. (1982)... Sekiguchi.G.J.. Tokyo 1:289-292. van Langen. pp..A. P. Balkema. P..[34] [35] Schanz. pp 2469-2478. (1999). and Adv. Vermeer. Proceedings of the 9th ICSMFE. Rheological characteristics of clays. China... P. and numerical stability. 693-709.. H. In: Grundbautaschenbuch Part 4. (1977). Properties of Soil and Rock (in German). Soil collapse computations with finite elements. Beyond 2000 in Computational Geotechnics.. John Wiley & Sons. P. Conf. ASCE Journal of the Geotechnical Engineering Division. (1990). No...E. Bonnier. A soft soil model and experiences with two integration schemes. R. (1977). Vermeer. Brinkgreve. Vaid. T.G. T.A. Wuhan..M. von Soos. Bonnier. Stolle. Vermeer. Special issue on Pre-failure deformation behaviour of geomaterials. H. From the classical theory of secondary compression to modern creep analysis.A. U. P. 3. Stolle. Numerical Models in Geomechanics. Edition 4. D. Programming the Finite Element Method. Chisester. R.F. (1991). An interpretation of initial stress and strain methods. (1989). Formulation and verification of the Hardening-Soil Model. 383-387. Comp. Schanz.F. Campanella. Geomech.G.E. Stolle. 9th Int. Griffith. Géotechnique 48. P. Concrete and Rock.K. D.F.V.. 4. Vermeer. . negative for compression Volumetric strain. positive for pressure Pre overburden pressure Equivalent shear stress or deviatoric stress Failure ratio Time Vector with displacement components Volumetric weight Dip angle Dip direction Increment Vector with Cartesian strain components. Su De e E Eoed f g G K K0 m M n OCR p pp POP q Rf t u γ α1 α2 ∆ ε : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cohesion Current stiffness parameter Undrained shear-strength Elastic material matrix representing Hooke's law Void ratio Young's modulus Oedometer modulus Yield function Plastic potential function Shear modulus Bulk modulus Coefficient of lateral earth pressure Power in stress-dependent stiffness relation Slope of critical state line in p'-q space Porosity Overconsolidation ratio Isotropic stress or mean stress. negative for pressure Vertical preconsolidation stress. negative for compression. normal components positive for tension. positive for pressure. normal components positive for extension.APPENDIX A SYMBOLS c csp Cu . negative for tension Isotropic preconsolidation stress. negative for pressure εv κ κ* λ λ λ* µ* ν σ σ . positive for extension Cam-Clay swelling index Modified swelling index Plastic multiplier Cam-Clay compression index Modified compression index Modified creep index Poisson's ratio Vector with Cartesian stress components. ϕ ψ : : Friction angle Dilatancy angle . K ): To copy K values from double array R1 to R2 MulVec( V. id2. im. R2. VecR ): To add n terms of two vectors. I2. V ): To initialize K terms of double array R to V SetIVal( I. idR : first dimension of matrices nR1 number of rows in xMat1 and resulting xMatR nC2 number of column in xMat2 and resulting xMatR nC1 number of columns in xMat2 =rows in xMat2 . Vec. n. a listing is given of the subroutines and functions which are provided by PLAXIS in libraries and source cod in the user-defined soil models directory. First dimension of matrix is im. VecR ): Matrix (xMat)-vector(Vec) operation. nR1. id2. n ): To multiply a vector V by a factor F. nC1. K. K ): To initialize K terms of double array R to zero MZeroI( I. result in VecR VecRi = R1 ⋅ Vec1i + R2 ⋅ Vec 2i MatMat( xMat1. idR ): Matrix multiplication xMatRij = xMat 1ik ⋅ xMat 2kj id1.APPENDIX B FORTRAN MODELS SUBROUTINES FOR USER-DEFINED SOIL In this appendix. IV ): To initialize K terms of integer array I to IV CopyIVec( I1. nC2. R2. K ): To initialize K terms of integer array I to zero SetRVal( R. R1. n. id1. F. K. xMat2. These can be called by the User_Mod subroutine: Subroutines MZeroR( R. resulting vector is VecR AddVec( Vec1. xMatR. n values MatVec( xMat. K ): To copy K values from integer array I1 to I2 CopyRVec( R1. Vec2. xN2. V. iV. PrnSig( iOpt. S2. S1. iOpt = 0 to obtain principal stresses without directions iOpt = 1 to obtain principal stresses and directions S array containing 6 stress components (XX. WriIVl( io. nr and nc are the number of rows and columns to print respectively. C. xN3 arrays containing principal directions (from PrnSig) . YY. B. S3 principal stresses xN1. S2. S1. ZZ. B contains inverse matrix of AOrig. S3. V. xN1. xN2. nd. I ): As WriVal but for integer value I WriVec( io. S3 sorted principal stresses ( S ≤ S2 ≤ S3 ) P isotropic stress (negative for compression) Q deviatoric stress CarSig( S1. nc ): As WriVal but for double matrix V. Q ): To determine principal stresses and (for iOpt=1) principal directions. C. n ): Matrix inversion of square matrices AOrig and B with dimensions n. S2. SNew ): To calculate Cartesian stresses from principal stresses and principal directions. YZ. S3. Row-pivotting is used. n ): As WriVal but for n values of integer array iV WriMat( io. xN3 array containing 3 values of principal normalized directions only when iOpt=1. AOrig is NOT destroyed. S1. C. xN1. xN3. P. nr. C. xN3. V ): To write a double value V to file unit io (when io > 0) The value is preceeded by the character string C. nd is first dimension of V. XY. S2.Matrix multiplication xMatRij = xMat 1ik ⋅ xMat 2kj Fully filled square matrices with dimensions n MatInvPiv( AOrig. xN2. C. ZX) xN1. xN2. WriVal( io. S. n ): As WriVal but for n values of double array V WriIVc( io. FALSE otherwise Logical Function Is0IArr( IArr. n ): Returns TRUE when all n values of real (double) array A are zero. B. n ): Returns the inproduct of two vectors with length n . xL ): To multiply the n components of vector xN such that the length of xN becomes xL (for example to normalize vector xN to unit length. Functions Logical Function LEqual( A. FALSE otherwise. n. XY.SNew contains 6 stress components (XX. ZZ. xN3 ): Cross product of vectors xN1 and xN2 SetVecLen( xN. FALSE otherwise Double Precision Function DInProd( A. xN2. LEqual = |A-B| < Eps * ( |A| + |B| + Eps ) / 2 Logical Function Is0Arr( A. YZ. YY. n ): Returns TRUE when all n values of integer array IArr are zero. ZX) CrossProd( xN1. Eps ): Returns TRUE when two values A and B are almost equal. B. . . The project directory is passed to the user subroutine by means of the parameters iPrjDir and iPrjLen. Logical IsOpen Inquire( unit = 1. If not. Here suggestions are given on how the debug file can be created and opened: 1: Inquire if a unit number is opened. This means that any IO unit number can be used without conflicts between the debug file and existing files used by PLAXIS.TRUE. The project directory string will always end with character 92 (\). End If The above suggestions assume that the debug file is located in the currently active directory.APPENDIX C CREATING A DEBUG-FILE FOR USER-DEFINED SOIL MODELS A debug file is not automatically created and opened in PLAXIS. Opened = IsOpen) If (..e. It is suggested that the debug file is stored in the DTA-directory of the corresponding PLAXIS project. The user has to rebuild the character string and can directly add the actual name of the debug file. which is not necessarily the proper location. Since the user subroutine is used many times. 255). ' ) IsOpen = . This is to avoid character passing conflicts (Fortran . File = ' . IsOpen) Then Open( Unit = 1. When writing a FORTRAN user subroutine and compiling it as a DLL. Therefore it is necessary to include also the pathname in the File = ' ... i.FALSE. '.C conflicts). The user should do this by including the corresponding source code in the user subroutine. The debug file needs only be created and opened once. IsOpen) Then Open( Unit = 1. open it. / Save IsOpen If (. ' ) End If 2: Use a DATA statement Logical IsOpen Data IsOpen / . The iPrjDir array contains the ASCII numbers of the characters of the project directory string and iPrjLen is the length of the string (max. files are not shared with the main program. The example underneath shows how a debug file called 'usrdbg' can be created and opened in the current project directory for debugging purposes: Character fName*255 . it must be checked whether the file.. the corresponding IO unit number. File = ' . is already open.Not..Not. / Save IsOpen If (. 6. Call WriVec( io.Not. 'Sig0'. 6 ) The available writing routines do not write when io is zero or negative. 6. End If In the user subroutine. 6 ) Call WriMat( io. 6 ) Call WriVec( io. 'Iter'. In any case. 'D'. Sig.1 . iStep. 1 x 1 unit model of a one-dimensional compression test or a triaxial test with zero soil weight)..TRUE. it is very useful to start by testing it with a simple finite element model in which a homogeneous stress state should occur (for example an axisymmetric. Int. 'Step'. Sig0..10 ) io = 1 . iPrjLen fName(i:i) = Char( iPrjDir(i) ) End Do fName = fName(:iPrjLen) // 'usrdbg' Open( Unit = 1. 'dEps'. File = fName ) IsOpen = . in order to avoid large debug files. IsOpen) Then fName = ' ' Do i=1.Eq. but the stress state in each point should be the same. Call WriIVl( io. 'Sig'.Eq. 1 ) Then . dEps. Debugging hints When developing and debugging a constitutive soil model in the user subroutine.. values can be written to IO unit 1. iStep ) Call WriIVl( io. The finite element model will still contain many stress points. using for example the available writing subroutines in Appendix B. 1 .And.And. Alternatively: If ( io .Eq. D. iTer ) Call WriVec( io. it is useful to write output for a limited number of stress points only (or for certain step numbers or iteration numbers).FALSE. Examples of writing useful but limited debug information are given below: io = 0 If ( iEl . 6 ) .Gt.Logical IsOpen Data IsOpen / .. When the file has not been opened before.j=1. This file must be opened before. Digital Fortran will open a file with the name FORT.2 in the current directory or checks the environment variable FORT2. * ) 'StVar:'. Lahey Fortran will give a Run-Time Error (File not open).2) End If Note that here file 2 is used.Write( 2.(StVar(j). . however.