Soil Behaviour and Geotechnical Modelling

(a) argue advantages and limitations of Dun fire and Changs stumper.Dun weed and Changs molding assumes a hyperbolic variant- hear relation and was developed based on triaxial soil tests. The original model assumes a constant Poissons balance magical spell the revised model accommodates the variation of Poissons ratio by means of deform- wagerent Poissons ratio or strive-dependent bulk modulus.The Dun endure-Chang model is advantageous in analyzing many working problems and is simple to set up with standard triaxial compression tests. When tri-axial test results ar non available, model parameters are also abundantly available in literatures. It is a simple yet obvious enhancement to the Mohr-Coulomb model. In this respect, this model is preferred over the Mohr-Coulomb model.However, it has its limitations, including, (i) the intermediate principal stress s2 is not accounted for (ii) results whitethorn be unreliable when extensive failure occurs (iii) it does not consider the vividness change referable to changes in prune stress (shear dilatancy) (iv) input parameters are not fundamental soil properties, however only empirical set for limited range of conditions. (v) the model is mainly intended for quasi-static analysis.(b) Discuss advantages and limitations of Yin and Grahams KGJ model.Yin and Grahams KGJ model is formed using data from isotropic consolidation tests and consolidated undrained triaxial tests with pore- irrigate pressure measurement. It put ups mouldal expressions for , , , and consanguinitys in soils.In Duncan and Changs model for triaxial stress conditionswhitethorn cause volume strain ( dilation and compression) may cause shear strain.Whereas Yin and Grahams KGJ modelThus the volume change and shear strain was interpreted into account, which is an improvement to Duncan and Changs model. The limitation of Yin and Grahams KGJ model may exist in the ratiocination of the parameter and the complexity of its calculation.(c) Dis cuss the differences amidst expansible models and hypo- waxy models.For soils, the behaviour depend on the stress path followed. The total contortion of such worldlys can be decomposed into a recoverable part and an irrecoverable part. Hypoelasticity constitutes a reason additive law in which the behaviour can be phoney from increment to increment rather than for the entire load or stress at a time. In hypoelasticity, the increment of stress is expressed as a function of stress and increment of strain. The Hypoelastic concept can provide simulation of constitutional behaviour in a smooth vogue and hence can be used for indurate or soften soils.Hypoelastic models can be considered as modification of linear elastic models. However, it may additively reversible, with no coupling between volumetric and deviatoric responses and is path-independent.5.2 Use sketches to explain the bodily (geometric) meaning of all 7 parameters (only 5 independent) in a cross-anisotropic elast ic soil model (). enroll 5.1 Parameters in cross-anisotropic elastic model Youngs modulus in the affirmational acresment Youngs modulus in the plane of deposition Poissons ratio for hard in the plane of deposition due to the stress acting in the direction of deposition Poissons ratio for straining in the direction of deposition due to the stress acting in the plane of deposition Poissons ratio for straining in the plane of deposition due to the stress acting in the same plane Shear modulus in the plane of the direction of deposition Shear modulus in the plane of deposition.Due to symmetry requirements, only 5 parameters are independent.Assignment 6 (Lecture 6 Elasto- moldable behaviour)6.1(a) Explain and discuss (i) chip in, (ii) topic criterion, (iii) potential come in, (iv) proceed territory, (v) atomic number 7, (vi) eubstance condition.(i) The give up strength or conk out acme of a material is defined in engineering and materials intelligence as the stress at w hich a material begins to deform credit cardally. precedent to the yield point the material will deform elastically and will heel counter to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the deformation will be permanent and non-reversible. In the uniaxial situations the yield stress indicates the onset of plastic straining. In the multi-axial situation it is not sensible to talk about a yield stress. Instead, a yield function is defined which is a scalar function of stress and pronounce parameters.(ii) A yield criterion, often expressed as yield surface, or yield locus, is an hypothesis concerning the limit of elasticity under any cabal of stresses. There are two interpretations of yield criterion one is purely mathematical in taking a statistical approach temporary hookup other models attempt to provide a justification based on established physical principles. Since stress and strain are tensor qualities they can be d escribed on the basis of three principal directions, in the cheek of stress these are denoted by , and .(iii) Potential surface is the segment of a plastic potential surface plotted in principal stress space, as shown in Figure 6.1 (a). A two dimensional bailiwick was shown in Figure 6.1 (b).(iv) Flow master a scalar multiplier plastic potential function location of surface (a vector), not in the final compareFigure 6.1 Plastic potential presentation(v) assumptive the plastic potential function to be the same as the yield function as a further simplificationThe incremental plastic strain vector is thence normal to the yield surface and the normality condition is verbalize to apply.(vi) Having defined the basic ingredients of an elasto-plastic constitutive model, a relationship between incremental stresses and incremental strains then can be obtained. When the material is plastic the stress state must satisfy the yield function. Consequently, on using the filament rule of di fferentiation, givesThis equation is known as the consistency equation or consistency condition.(b) Explain and discuss the associate flow rule and non-associate flow rule and how the two rules affect the volumetric deformation and the bearing capacity of a strip footing on sand.sometimes simplification can be applied by assuming the plastic potential function to be the same as the yield function (i.e. ). In this case the flow rule is said to be associated. The incremental plastic strain vector is then normal to the yield surface and the normality condition is said to apply. In the general case in which the yield and plastic potential functions differ (i.e. ), the flow rule is said to be non-associated. If the flow rule is associated, the constitutive hyaloplasm is even and so is the global stiffness matrix. On the other hand, if the flow rule is non-associated twain the constitutive matrix and the global stiffness matrix become non-symmetric. The eversion of non-symmetric matri ces is much more costly, both of storage and computer time.As noted, it occurs in a special class of plasticity in which the flow rule is said to be associated. Substitution of a symmetric for all atoms in a finite element mesa, into the assembly process, results in a symmetric global stiffness matrix. For the general case in which the flow rule is non-associated and the yield and plastic potential functions differ, the constitutive matrix is non-symmetric. When assembled into the finite element equations this results in a non- symmetric global stiffness matrix. The inversion of such a matrix is more complex and requires more computing resources, both memory and time, than a symmetric matrix. Some commercial programs are ineffective to deal with non-symmetric global stiffness matrices and, consequently, restrict the typo of plastic models that can be accommodated to those which have an associated flow rule.(c) Explain plastic strain harden and plastic work hardening or softening. The state parameters, , are link to the accumulated plastic strains . Consequently, if there is a linear relationship between and so thatthen on substitution, along with the flow rule, the unknown scalar,, cancels and A becomes determinant. If there is not a linear relationship between and , the first derivative ratio on the left hand side of the above equation is a function the plastic strains and therefore a function of . When substituted, along with the flow rule given, the As do not cancel and A becomes indeterminate. It is then not possums to evaluate the . In practice all strain hardening/softening models assume a linear relationship between the state parameters and the plastic strains .In this type of plasticity the state parameters, are relate to the accumulated plastic work, ,which is dependent on the plastic strains it can be shown, following a similar argument to that parented above for strain hardening/softening plasticity, that as long as there is a linear relationsh ip between the state parameters , and the plastic work, , the parameter defined becomes independent of the unknown scalar, , send therefore is determinant. If the relationship between and is not linear, become a function of and it is not possible to evaluate the constitutive matrix.6.2 Show steps to derive the elastic plastic constitutive matrix in (6.16).The incremental total strains can be split into elastic and plastic , componets. The incremental stress, are related to the incremental elastic strains, by the elastic constitutive matrixOr alternativelyCombining givesThe incremental plastic strains are related to the plastic potential function, via the flow rule. This can be written asSubstituting givesWhen the material is plastic the stress state must satisfy the yield function. Consequently, which, on using the chain rule of differentiation.This equation is known as the consistency equation. It can be rearranged to giveCombining, we can getWhereSubstituting againSo that6.3 The dimension of a slope is shown in Figure 6.2. Calculate the factor of safety device of the following cases(a) Without strain crack, the properties of Soil (1) are kPa, , kN/m3 The properties of Soil (2) are kPa, , kN/m3 (no peeing table).(b) With latent hostility crack filled with irrigate supply, repeat the calculation in (a).(c) Without tension crack, the properties of Soil (1) are kPa, , kN/m3 (below wet table) and kN/m3 (above water table) the properties of Soil (2) are kPa, , kN/m3 (below water table) and kN/m3 (above water table). Water table is shown.Figure 6.2 Dimension of the slope and water table(a)Figure 6.3 Model without tension crack or water tableFactor of Safety 1.498Figure 6.4 Results without tension crack or water tableFigure 6.5 patch 1 Morgenstern-Price method(b)Figure 6.6 Model with tension crack filled with waterFigure 6.7 Results with tension crack filled with waterThe safety factor 1.406Figure 6.8 Slice 1 Morgenstern-Price Method(c)Figure 6.9 Model without tension crack but with water tableFigure 6.10 Results without tension crack but with water tableFactor of Safety 1.258Figure 6.11 Slice 1 Morgenstern-Price Method